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A '''statistical model''' is a formalization of relationships between variables in the form of mathematical equations. A statistical model describes how one or more [[random variables]] are related to one or more other variables. The model is statistical as the variables are not [[Deterministic system|deterministically]] but [[stochastic]]ally related. In mathematical terms, a statistical model is frequently thought of as a pair <math> (Y, P) </math> where <math> Y </math> is the set of possible observations and <math> P </math> the set of possible [[probability distributions]] on <math> Y </math>. It is assumed that there is a distinct element of <math> P </math> which generates the observed data. [[Statistical inference]] enables us to make statements about which element(s) of this set are likely to be the true one.
On Sunday, Nike hosted its annual half-marathon through London, Run to the Beat, with more than 19,000 runners pounding around the city.<br>For the past two years, the event's "hydration partner" - the sponsor that hands free drinks to the athletes as they run - has been Powerade, an isotonic sports drink, with its bold graphics and intimations of muscle-making.<br><br>


Most statistical tests can be described in the form of a statistical model. For example, the [[Student's t-test]] for comparing the means of two groups can be formulated as seeing if an estimated [[parameter]] in the model is different from 0. Another similarity between tests and models is that there are assumptions involved. Error is assumed to be normally distributed in most models.<ref>Field, A. (2005). Discovering statistics using SPSS. Sage, London.</ref>
This year, the runners were drinking coconut water. Goodbye machismo endurance, hello palm trees: and it didn't seem in the slightest bit [http://Www.Squidoo.com/search/results?q=strange strange].<br>That's because it's more than just a clever marketing deal - it's a symbol of a significant shift in the way we think about a drink that here in the UK we couldn't even buy before 2010.<br><br>Three years after we were introduced to the exotic idea, Pret A Manger now stocks its own brand of coconut water, and last week, Waitrose reported a rise of 183 per cent in the sale of coconut water and oil. How did something so niche get so mainstream?<br>Of course, what the Europeans and Americans consider a new trend usually started somewhere else. This one, coconut water, was pretty much the preserve of Brazilians and anyone else living near palm trees who could manage to poke a drinking straw through the shell.<br><br>Different to coconut milk, the water comes from young coconuts, is clear rather than creamy, and was used for emergency plasma transfusions during the Second World War. The Indians call it "miracle water". It's not a new discovery. But maybe all those people overlooked its market value.<br><br>At any rate, they didn't do the tests that the new wave of sales-pitch-ready salesmen have done.<br><br>The tests revealed that coconut water contains more potassium than a banana, has naturally occurring electrolytes which can replace those we lose through sweat when exercising, has no fat, and no cholesterol.<br>In 2004, when Vita Coco and Zico launched their branded coconut waters in the United States, they quickly found that yoga fans and sporty types liked the first two properties, and celebrities liked them all. In a very natural trajectory, Madonna invested almost $1.5m in Vita Coco in 2010, Demi Moore and Matthew McConaughey came on board to buy stakes in the brand, too, and coconut water seemed to arc effortlessly from humble origins to the preserve of the A-listers.<br><br>It makes perfect sense that Rihanna is the face of Vita Coco in the US. Huge billboards show her sipping a carton of Vita Coco on a beach. It's not a great leap to imagine that her image was what finally brought coconut water out of the yoga class and into the grit of real life.<br>People began to say how much it helped hangovers. Unofficial celebrity spokespeople helped, too, because they were famous and they looked good holding it: a quick internet search will find any number of them (Gwyneth Paltrow, Sienna Miller, Jessica Simpson et al) papped with a container of coconut water.<br><br>Naysayers claim that coconut water is just another passing fad, but the figures suggest otherwise. In 2011, there were six brands in the UK. Today, there are 36. Vita Coco, the leading brand, had a market share of 76 per cent in 2011. Now it has 91 per cent, which is commensurate with the fact that, since 2010, it has doubled its business each year.<br><br>Vita Coco is a privately held company, so profits are not available, but what it calls "retail value sales" are substantial. This year, they will have reached almost �40m. Mintel predicts that the company will be worth �100m in the UK by 201<br>
Put this in context, and the figures are even more baffling: the awareness level of coconut water in the is less than 3 per cent of the population, and only 1 per cent of households buy it. "It's all about untapped potential," Giles Brook, the CEO of Vita Coco Europe, told The Independen<br><br>
Brook was on the board of Innocent Drinks until four years ago, so knows how an idea can work its way from health freaks' standard to standard item on the weekly shop. It's the humble made desirable. Key ingredient: aspiration. But it's no mean feat letting people know what their aspirations should <br><br>


==Formal definition==
Brook says that although Vita Coco's "big advertising campaign" kicked off yesterday - all beach and palm trees - the approach here is very much grassroo<br>.
"In the UK, it's about taking the brand out to consumers, not just dropping the brand onto a billboard," he says. "We build the brand bottom up� lots of sampling," he says. If you've tasted coconut water, you might wonder whether this technique is  [http://www.futurecol.co.nz/assets/nike-online.asp nike free] perhaps about more than spreading the w<br><br><br>


A statistical model is a collection of [[Cumulative distribution function|probability distribution functions]] or [[probability density function]]s (collectively referred to as ''distributions'' for brevity). A [[parametric model]] is a collection of distributions, each of which is indexed by a unique finite-dimensional parameter: <math>\mathcal{P}=\{\mathbb{P}_{\theta} : \theta \in \Theta\}</math>, where <math>\theta</math> is a parameter and <math>\Theta \subseteq \mathbb{R}^d</math> is the feasible region of parameters, which is a subset of d-dimensional [[Euclidean space]].  A statistical model may be used to describe the set of distributions from which one assumes that a particular data set is sampled.  For example, if one assumes that data arise from a univariate [[Gaussian distribution]], then one has assumed a Gaussian model
Let's just say, coconut water can be an acquired taste. When the staff of the Huffington Post did a blind taste test of 12 brands of coconut water earlier this summer, comments included: "Tastes like rotten old plant water that results after leaving your flowers in a vase for way too long", "something's off", "Dear Lord murder me" and "inspired my gag refl<br><br><br>
:<math>\mathcal{P}=\{\mathbb{P}(x; \mu, \sigma) = \frac{1}{\sqrt{2 \pi} \sigma} \exp\left\{ -\frac{1}{2\sigma^2}(x-\mu)^2\right\} : \mu \in \mathbb{R}, \sigma > 0\}</math>.


A [[non-parametric model]] is a set of probability distributions with infinite dimensional parameters, and might be written as <math>\mathcal{P}=\{\text{all distributions}\}</math>.  A [[semi-parametric model]] also has infinite dimensional parameters, but is not dense in the space of distributions.  For example, a mixture of Gaussians with one Gaussian at each data point is dense in the space of distributions. Formally, if d is the dimension of the parameter, and n is the number of samples, if <math>d \rightarrow \infty</math> as <math>n \rightarrow \infty</math> and <math>d/n \rightarrow 0</math> as <math>n \rightarrow \infty</math>, then the model is semi-parametric.
Such views were slightly more engaging than the good comments - Vita's version drew "fresh tasting", "palatable", "mild, pleasant". Brook laughs about these initial reactions, explaining that if people don't have great memories about the first taste, they often like it more second time aro<br><br>


==Model comparison==
"It grows on peop<br><br>
Models can be compared to each other. This can either be done when you have done an [[exploratory data analysis]] or a [[confirmatory data analysis]]. In an exploratory analysis, you formulate all models you can think of, and see which describes your data best. In a confirmatory analysis you test which of your models you have described before the data was collected fits the data best, or test if your only model fits the data. In linear regression analysis you can compare the amount of variance explained by the independent variables, R<sup>2</sup>, across the different models. In general, you can compare models that are nested by using a [[Likelihood-ratio test]]. Nested models are models that can be obtained by restricting a parameter in a more complex model to be zero.


==An example==
Coconut water is the fastest growing category in non-alcoholic beverages in the UK, and which brand will come out on top is about survival of the fittest - Vita Coco is already in more than 10,000 outlets in the UK, and in around 16,000 across Europe; it's a lucrative busin<br>s.
Height and age are probabilistically distributed over humans. They are stochastically related; when you know that a person is of age 7, this influences the chance of this person being 6 feet tall. You could formalize this relationship in a [[linear regression model]] of the following form:
Is it lucrative for the growers, though? It should pay to be a coconut farmer these days - coconuts are more in demand than ever bef<br><br>
height<sub>i</sub> = b<sub>0</sub> + b<sub>1</sub>age<sub>i</sub> + ε<sub>i</sub>, where b<sub>0</sub> is the intercept, b<sub>1</sub> is a parameter that age is multiplied by to get a prediction of height, ε is the error term, and i is the subject. This means that height starts at some value, there is a minimum height when someone is born, and it is predicted by age to some amount. This prediction is not perfect as error is included in the model. This error contains variance that stems from sex and other variables. When sex is included in the model, the error term will become smaller, as you will have a better idea of the chance that a particular 16-year-old is 6 feet tall when you know this 16-year-old is a girl. The model would become height<sub>i</sub> = b<sub>0</sub> + b<sub>1</sub>age<sub>i</sub> + b<sub>2</sub>sex<sub>i</sub> + ε<sub>i</sub>, where the variable sex is [[dichotomous]]. This model would presumably have a higher R<sup>2</sup>. The first model is nested in the second model: the first model is obtained from the second when b<sub>2</sub> is restricted to zero.


==Classification==
O.N.E, a brand in which PepsiCo has the majority stake, ran out of Brazilian coconuts in 2011, and had to turn to the Philippines and Indonesia for supplies. It's hard to believe that companies are fiercely competitive over this strange-smelling liquid that was until recently considered a waste prod<br><br>
According to the number of the endogenous variables and the number of equations, models can be classified as complete models (the number of equations equal to the number of endogenous variables) and incomplete models. Some other statistical models are the [[general linear model]] (restricted to continuous dependent variables), the [[generalized linear model]] (for example, [[logistic regression]]), the [[multilevel model]], and the [[structural equation model]].<ref>[[Adèr, H.J.]] (2008). Chapter 12: Modelling. In [[H.J. Adèr]] & [[Gideon J. Mellenbergh|G.J. Mellenbergh]] (Eds.) (with contributions by D.J. Hand), Advising on Research Methods: A consultant's companion (pp. 271-304). Huizen, The Netherlands: Johannes van Kessel Publishing.</ref>


==See also==
It wasn't unusual for factories in Asia and Brazil to wait 14 months for a coconut to mature enough to harvest the coconut oil and the milk, and then dump rivers of the unwanted coconut water. It's a different world now, and an entirely different form of industry has been spaw<br><br>
{{Commons category|{{PAGENAME}}}}
*[[A/B testing]]
*[[Econometric model]]
*[[Mathematical diagram]]
*[[Regression analysis]]


{{More footnotes|date=September 2010}}
Pret A Manger's coconut water comes from a single plantation in Thailand; Zico - a brand in which Coca-Cola has a major stake, and which produces its drink from concentrate - says that it is "continuously searching the world for the best-tasting and most nutritious coconu<br><br>


==References==
It's coconuts are from Thailand, Indonesia and the Philippines. However, the company doesn't give away much about the deals it strikes with the farm<br><br>
<references/>


{{DEFAULTSORT:Statistical Model}}
Harmless Harvest, a brand that creates a raw organic version of the drink, practices agro-forestry. Its promise of "constructive capitalism" was tested this summer: on its Facebook page last month, fans in the US were desperate to find out why the product had disappeared from the shel<br><br>
[[Category:Statistical models| ]]
 
[[Category:Statistical theory]]
There were, says co-founder Justin Guilbert, "challenges in building up manufacturing capabilities with a safety stock that got gobbled up by surging demand". But unlike some of Harmless Harvest's rivals, when its source in Thailand couldn't provide as many coconuts as were needed, the company didn't move<br><br>
[[Category:Scientific modeling]]
 
"We chose to invest in communities and at the source," Guilbert says. "We also chose not to compromise. If we did, consumers would lose all interest. The beauty of an ecosystem-based business model is that you're only as good as the way you treat the other agents, from plant source to consumer." He thinks the reason that none of the retailers or consumers abandoned the brand in the brief drought is "the result of not taking people on a marketing ride" and investing in the product inst<br><br><br>
 
His hope is that Harmless Harvest's way will "reach a critical mass" and agri-business will eventually be seen as a competitive option. "We launched a couple of years ago in a world of aseptic concentrate big business or venture-capital-owned companies. Against all, we decided to build a supply chain from scratch� The result has been astounding." The company is now one of the fastest-growing in the American food and beverage se<br><br>.
 
Vita Coco also talks about ethics: it says that it harvests from seven countries and works directly with farmers. Its corporate social responsibility, the company says, is shown by how it uses the coconut: once the water is extracted, the flesh is used for food products, and the husk used for fibre in products or as a renewable energy so<br><br><br>
 
 
It is startling what a trend, and a hefty amount of investment, can do - it can alter a country's industry, and nudge the workings of an ecosystem into an entirely different direction. And just like that, drinking pure coconut water a million miles away from a palm tree seems like the most natural thing in the w<br><br>.
 
Video: Rihanna on her new collection with River Island

Latest revision as of 17:40, 8 December 2014

On Sunday, Nike hosted its annual half-marathon through London, Run to the Beat, with more than 19,000 runners pounding around the city.
For the past two years, the event's "hydration partner" - the sponsor that hands free drinks to the athletes as they run - has been Powerade, an isotonic sports drink, with its bold graphics and intimations of muscle-making.

This year, the runners were drinking coconut water. Goodbye machismo endurance, hello palm trees: and it didn't seem in the slightest bit strange.
That's because it's more than just a clever marketing deal - it's a symbol of a significant shift in the way we think about a drink that here in the UK we couldn't even buy before 2010.

Three years after we were introduced to the exotic idea, Pret A Manger now stocks its own brand of coconut water, and last week, Waitrose reported a rise of 183 per cent in the sale of coconut water and oil. How did something so niche get so mainstream?
Of course, what the Europeans and Americans consider a new trend usually started somewhere else. This one, coconut water, was pretty much the preserve of Brazilians and anyone else living near palm trees who could manage to poke a drinking straw through the shell.

Different to coconut milk, the water comes from young coconuts, is clear rather than creamy, and was used for emergency plasma transfusions during the Second World War. The Indians call it "miracle water". It's not a new discovery. But maybe all those people overlooked its market value.

At any rate, they didn't do the tests that the new wave of sales-pitch-ready salesmen have done.

The tests revealed that coconut water contains more potassium than a banana, has naturally occurring electrolytes which can replace those we lose through sweat when exercising, has no fat, and no cholesterol.
In 2004, when Vita Coco and Zico launched their branded coconut waters in the United States, they quickly found that yoga fans and sporty types liked the first two properties, and celebrities liked them all. In a very natural trajectory, Madonna invested almost $1.5m in Vita Coco in 2010, Demi Moore and Matthew McConaughey came on board to buy stakes in the brand, too, and coconut water seemed to arc effortlessly from humble origins to the preserve of the A-listers.

It makes perfect sense that Rihanna is the face of Vita Coco in the US. Huge billboards show her sipping a carton of Vita Coco on a beach. It's not a great leap to imagine that her image was what finally brought coconut water out of the yoga class and into the grit of real life.
People began to say how much it helped hangovers. Unofficial celebrity spokespeople helped, too, because they were famous and they looked good holding it: a quick internet search will find any number of them (Gwyneth Paltrow, Sienna Miller, Jessica Simpson et al) papped with a container of coconut water.

Naysayers claim that coconut water is just another passing fad, but the figures suggest otherwise. In 2011, there were six brands in the UK. Today, there are 36. Vita Coco, the leading brand, had a market share of 76 per cent in 2011. Now it has 91 per cent, which is commensurate with the fact that, since 2010, it has doubled its business each year.

Vita Coco is a privately held company, so profits are not available, but what it calls "retail value sales" are substantial. This year, they will have reached almost �40m. Mintel predicts that the company will be worth �100m in the UK by 201
Put this in context, and the figures are even more baffling: the awareness level of coconut water in the is less than 3 per cent of the population, and only 1 per cent of households buy it. "It's all about untapped potential," Giles Brook, the CEO of Vita Coco Europe, told The Independen

Brook was on the board of Innocent Drinks until four years ago, so knows how an idea can work its way from health freaks' standard to standard item on the weekly shop. It's the humble made desirable. Key ingredient: aspiration. But it's no mean feat letting people know what their aspirations should

Brook says that although Vita Coco's "big advertising campaign" kicked off yesterday - all beach and palm trees - the approach here is very much grassroo
. "In the UK, it's about taking the brand out to consumers, not just dropping the brand onto a billboard," he says. "We build the brand bottom up� lots of sampling," he says. If you've tasted coconut water, you might wonder whether this technique is nike free perhaps about more than spreading the w


Let's just say, coconut water can be an acquired taste. When the staff of the Huffington Post did a blind taste test of 12 brands of coconut water earlier this summer, comments included: "Tastes like rotten old plant water that results after leaving your flowers in a vase for way too long", "something's off", "Dear Lord murder me" and "inspired my gag refl


Such views were slightly more engaging than the good comments - Vita's version drew "fresh tasting", "palatable", "mild, pleasant". Brook laughs about these initial reactions, explaining that if people don't have great memories about the first taste, they often like it more second time aro

"It grows on peop

Coconut water is the fastest growing category in non-alcoholic beverages in the UK, and which brand will come out on top is about survival of the fittest - Vita Coco is already in more than 10,000 outlets in the UK, and in around 16,000 across Europe; it's a lucrative busin
s. Is it lucrative for the growers, though? It should pay to be a coconut farmer these days - coconuts are more in demand than ever bef

O.N.E, a brand in which PepsiCo has the majority stake, ran out of Brazilian coconuts in 2011, and had to turn to the Philippines and Indonesia for supplies. It's hard to believe that companies are fiercely competitive over this strange-smelling liquid that was until recently considered a waste prod

It wasn't unusual for factories in Asia and Brazil to wait 14 months for a coconut to mature enough to harvest the coconut oil and the milk, and then dump rivers of the unwanted coconut water. It's a different world now, and an entirely different form of industry has been spaw

Pret A Manger's coconut water comes from a single plantation in Thailand; Zico - a brand in which Coca-Cola has a major stake, and which produces its drink from concentrate - says that it is "continuously searching the world for the best-tasting and most nutritious coconu

It's coconuts are from Thailand, Indonesia and the Philippines. However, the company doesn't give away much about the deals it strikes with the farm

Harmless Harvest, a brand that creates a raw organic version of the drink, practices agro-forestry. Its promise of "constructive capitalism" was tested this summer: on its Facebook page last month, fans in the US were desperate to find out why the product had disappeared from the shel

There were, says co-founder Justin Guilbert, "challenges in building up manufacturing capabilities with a safety stock that got gobbled up by surging demand". But unlike some of Harmless Harvest's rivals, when its source in Thailand couldn't provide as many coconuts as were needed, the company didn't move

"We chose to invest in communities and at the source," Guilbert says. "We also chose not to compromise. If we did, consumers would lose all interest. The beauty of an ecosystem-based business model is that you're only as good as the way you treat the other agents, from plant source to consumer." He thinks the reason that none of the retailers or consumers abandoned the brand in the brief drought is "the result of not taking people on a marketing ride" and investing in the product inst


His hope is that Harmless Harvest's way will "reach a critical mass" and agri-business will eventually be seen as a competitive option. "We launched a couple of years ago in a world of aseptic concentrate big business or venture-capital-owned companies. Against all, we decided to build a supply chain from scratch� The result has been astounding." The company is now one of the fastest-growing in the American food and beverage se

.

Vita Coco also talks about ethics: it says that it harvests from seven countries and works directly with farmers. Its corporate social responsibility, the company says, is shown by how it uses the coconut: once the water is extracted, the flesh is used for food products, and the husk used for fibre in products or as a renewable energy so



It is startling what a trend, and a hefty amount of investment, can do - it can alter a country's industry, and nudge the workings of an ecosystem into an entirely different direction. And just like that, drinking pure coconut water a million miles away from a palm tree seems like the most natural thing in the w

.

Video: Rihanna on her new collection with River Island