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on 16-Sep-2019 (Mon)

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Flashcard 4388927966476

Question
To work our way up to GLMs, we will begin by defining exponential family distributions. We say that a class of distributions is in the exponential family if it can be written in the form
Answer

\(p(y ; \eta)=b(y) \exp \left(\eta^{T} T(y)-a(\eta)\right)\)

Here, \(\eta\) is called the natural parameter (also called the canonical parameter) of the distribution; T(y) is the sufficient statistic (for the distributions we consider, it will often be the case that T(y) = y); and \(a(\eta)\) is the log partition function. The quantity \(e^{-a(\eta)}\) essentially plays the role of a normalization constant, that makes sure the distribution \(p(y;\eta)\) sums/integrates over y to 1.


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Flashcard 4388930325772

Question

To work our way up to GLMs, we will begin by defining exponential family distributions. We say that a class of distributions is in the exponential family if it can be written in the form

\(p(y ; \eta)=b(y) \exp \left(\eta^{T} T(y)-a(\eta)\right)\)

Here, \(\eta\) is called the [...] (also called the [...] ) of the distribution; T(y) is the sufficient statistic (for the distributions we consider, it will often be the case that T(y) = y); and \(a(\eta)\) is the log partition function. The quantity \(e^{-a(\eta)}\) essentially plays the role of a normalization constant, that makes sure the distribution \(p(y;\eta)\) sums/integrates over y to 1.

Answer

natural parameter

canonical parameter


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Flashcard 4388932160780

Question

To work our way up to GLMs, we will begin by defining exponential family distributions. We say that a class of distributions is in the exponential family if it can be written in the form

\(p(y ; \eta)=b(y) \exp \left(\eta^{T} T(y)-a(\eta)\right)\)

Here, \(\eta\) is called the natural parameter (also called the canonical parameter) of the distribution; T(y) is the [...] (for the distributions we consider, it will often be the case that T(y) = y); and \(a(\eta)\) is the log partition function. The quantity \(e^{-a(\eta)}\) essentially plays the role of a normalization constant, that makes sure the distribution \(p(y;\eta)\) sums/integrates over y to 1.

Answer

sufficient statistic


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Flashcard 4388933995788

Question

To work our way up to GLMs, we will begin by defining exponential family distributions. We say that a class of distributions is in the exponential family if it can be written in the form

\(p(y ; \eta)=b(y) \exp \left(\eta^{T} T(y)-a(\eta)\right)\)

Here, \(\eta\) is called the natural parameter (also called the canonical parameter) of the distribution; T(y) is the sufficient statistic (for the distributions we consider, it will often be the case that T(y) = y); and \(a(\eta)\) is the [...] . The quantity \(e^{-a(\eta)}\) essentially plays the role of a normalization constant, that makes sure the distribution \(p(y;\eta)\) sums/integrates over y to 1.

Answer
log partition function


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Flashcard 4388935830796

Tags
#machine-learning #statistics #supervised-learning
Question
In CS229, what 3 modelling assumptions are used to derive GLMs?
Answer

1. \(y | x ; \theta \sim \text { Exponential Family }(\eta)\)

2. Goal is to predict \(h(x)=\mathrm{E}[T(y) | x]\)

3. The natural parameter \(\eta\) and the inputs x are related linearly: \(\eta = \theta^Tx\)


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Flashcard 4388938190092

Question
To introduce a little more terminology, the function g giving the distribution’s mean as a function of the natural parameter (\(g(\eta)=\mathrm{E}[T(y) ; \eta]\)) is called the [...] function. Its inverse, g −1 , is called the [...] function.
Answer

canonical response

canonical link

E.g. \(\phi=g(\eta)=1 /\left(1+e^{-\eta}\right)\)is the canonical response function for the Bernoulli distribution, \(\eta= g^{-1}(\phi)=\log (\phi /(1-\phi))\) is the canonical link function.


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This paper draws on the findings of a field study car- ried out in the south-west of England over the winter of 2014/15.

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Importantly, there was no dedicated thermal storage (e.g. hot water tanks) beyond the thermal inertia of the building. This is likely to become the prevailing situation in the UK where many hot water tanks are being removed in favour of combi boilers (Palmer & Cooper, 2013).

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There are two common features of these modelling studies. Firstly, they all assume some sort of additional thermal storage is available beyond the thermal mass of the building fabric. This may not be a reasonable assumption as anecdotal evidence in the UK suggests that additional storage such as buffer tanks are not regularly installed alongside heat pumps. While buffer tanks are important for demand shifting, Green (2012) found the installation of buffer tanks delivered little or no improvement in heat pump performance in field test- ing of heat pumps under standard operating conditions. Buffer tanks are not mentioned in the UK heat pump- installer guidance (Microgeneration Certification Scheme, 2015) and, indeed, none of the field trial homes considered in this paper has buffer tanks.

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Also on this theme, the Danish eFlex project (Dong Energy, 2012) tested the ability of heat pumps to provide turn-down DSR in response to price or wind-generation signals (or a combination of both), while participants could choose to optimize for cost or comfort. Participants only overrode turn-down events (lasting 1–3h)1%ofthe time, or approximately once in three months, leading the report authors to conclude that ‘the customers’ comfort zone has not been seriously challenged’ (p. 36).

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The contribution of the current paper is to present empirical evidence on the effectiveness and acceptability of demand shifting where the heat pumps uses fabric inertia alone to enable DSR. Analysis of both monitored and survey and interview data gathered in the field study provides new insights in terms of both the technical per- formance and limitations of this approach and also, importantly, the perceptions of real UK customers.

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The HEMS runs a self-learning, predictive control algorithm that develops a thermal model of the home by observing the building’s behaviour in response to heat input, external weather conditions and other fac- tors. This thermal model is used to cost-optimize the operation of the participant’s heat pump. This means it should reach the specified target temperatures at the time periods set, but at lower cost than if the algorithm were not being applied. This is a complex, multidimen- sional trade-off as discussed by Carter, Lancaster, and Chanda (2017), who present modelled outputs from the algorithm.

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if X is a set of airports and xRy means "there is a direct flight from airport x to airport y" (for x and y in X), then the [...] of R on X is the relation R+ such that x R+ y means "it is possible to fly from x to y in one or more flights". Informally, the [...] gives you the set of all places you can get to from any starting place.

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Transitive closure - Wikipedia
, see transitive set § Transitive closure. In mathematics, the transitive closure of a binary relation R on a set X is the smallest relation on X that contains R and is transitive. For example, <span>if X is a set of airports and xRy means "there is a direct flight from airport x to airport y" (for x and y in X), then the transitive closure of R on X is the relation R+ such that x R+ y means "it is possible to fly from x to y in one or more flights". Informally, the transitive closure gives you the set of all places you can get to from any starting place. More formally, the transitive closure of a binary relation R on a set X is the transitive relation R+ on set X such that R+ contains R and R+ is minimal Lidl & Pilz (1998, p. 337).