Blue Spaces Health Benefits Assessment Methods

Blue spaces health benefits assessment methods

Assessing the environmental benefits of a blue space intervention can be done by conducting a Health impact assessment (HIA). Additionally, a group of researches has developed a novel tool specifically designed to quantify the health benefits of blue spaces.

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Feature vectors and feature spaces

In some applications, it is not sufficient to extract only one type of feature to obtain the relevant information from the image data. Instead two or more different features are extracted, resulting in two or more feature descriptors at each image point. A common practice is to organize the information provided by all these descriptors as the elements of one single vector, commonly referred to as a feature vector. The set of all possible feature vectors constitutes a feature space. A common example of feature vectors appears when each image point is to be classified as belonging to a specific class. Assuming that each image point has a corresponding feature vector based on a suitable set of features, meaning that each class is well separated in the corresponding feature space, the classification of each image point can be done using standard classification method. Another and related example occurs when neural network-based processing is applied to images. The input data fed to the neural network is often given in terms of a feature vector from each image point, where the vector is constructed from several different features extracted from the image data. During a learning phase, the network can itself find which combinations of different features are useful for solving the problem at hand.

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The p-norm in infinite dimensions and â„“p spaces

The sequence space pThe p-norm can be extended to vectors that have an infinite number of components (sequences), which yields the space p. This contains as special cases: 1, the space of sequences whose series is absolutely convergent, 2, the space of square-summable sequences, which is a Hilbert space, and , the space of bounded sequences.The space of sequences has a natural vector space structure by applying addition and scalar multiplication coordinate by coordinate. Explicitly, the vector sum and the scalar action for infinite sequences of real (or complex) numbers are given by: ( x 1 , x 2 , ... , x n , x n 1 , ... ) ( y 1 , y 2 , ... , y n , y n 1 , ... ) = ( x 1 y 1 , x 2 y 2 , ... , x n y n , x n 1 y n 1 , ... ) , ( x 1 , x 2 , ... , x n , x n 1 , ... ) = ( x 1 , x 2 , ... , x n , x n 1 , ... ) . displaystyle beginaligned&(x_1,x_2,ldots ,x_n,x_n1,ldots )(y_1,y_2,ldots ,y_n,y_n1,ldots )=&(x_1y_1,x_2y_2,ldots ,x_ny_n,x_n1y_n1,ldots ),pt]&lambda cdot left(x_1,x_2,ldots ,x_n,x_n1,ldots

ight)=&(lambda x_1,lambda x_2,ldots ,lambda x_n,lambda x_n1,ldots ).endaligned Define the p-norm: x p = ( | x 1 | p | x 2 | p | x n | p | x n 1 | p ) 1 / p displaystyle left|x

ight|_p=left(|x_1|^p|x_2|^pcdots |x_n|^p|x_n1|^pcdots

ight)^1/p Here, a complication arises, namely that the series on the right is not always convergent, so for example, the sequence made up of only ones, (1, 1, 1, ...), will have an infinite p-norm for 1 p 1, as the series 1 p 1 2 p 1 n p 1 ( n 1 ) p , displaystyle 1^pfrac 12^pcdots frac 1n^pfrac 1(n1)^pcdots , diverges for p = 1 (the harmonic series), but is convergent for p > 1. One also defines the -norm using the supremum: x = sup ( | x 1 | , | x 2 | , ... , | x n | , | x n 1 | , ... ) displaystyle left|x

ight|_infty =sup(|x_1|,|x_2|,dotsc ,|x_n|,|x_n1|,ldots ) and the corresponding space of all bounded sequences. It turns out that x = lim p x p displaystyle left|x

ight|_infty =lim _pto infty left|x

ight|_p if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider p spaces for 1 p . The p-norm thus defined on p is indeed a norm, and p together with this norm is a Banach space. The fully general Lp space is obtained-as seen below-by considering vectors, not only with finitely or countably-infinitely many components, but with "arbitrarily many components"; in other words, functions. An integral instead of a sum is used to define the p-norm. General p-spaceIn complete analogy to the preceding definition one can define the space p ( I ) displaystyle ell ^p(I) over a general index set I displaystyle I (and 1 p