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An important operation in signal processing and machine learning is dimensionality reduction. There are many such methods, but the starting point is usually linear methods that map data to a lower-dimensional set called a subspace. When working with matrices, the notion of dimension is quantified by rank. This chapter reviews subspaces, span, dimension, rank, and nullspace. These linear algebra concepts are crucial to thoroughly understanding the SVD, a primary tool for the rest of the book (and beyond). The chapter concludes with a machine learning application, signal classification by nearest subspace, that builds on all the concepts of the chapter.
Chapter 1: In this chapter, we provide formal definitions of real and complex vector spaces, and many examples. Among the important concepts introduced are linear combinations, span, linear independence, and linear dependence.
The study aimed to examine nutrition label use and dietary behaviours among ethnically diverse middle- and high-school students, in Texas, USA.
Design
The School Physical Activity and Nutrition (SPAN) survey is a cross-sectional statewide study using a self-administered questionnaire to assess nutrition and physical activity behaviours. Height and weight measurements were used to determine BMI. Multivariable logistic regression was used to determine associations between nutrition label use and dietary behaviours, with gender, grade, ethnicity, BMI, parent education, socio-economic status and nutrition knowledge as covariates.
Setting
Participants from 283 schools, weighted to represent Texas youth.
Subjects
SPAN 2009–2011 included 6716 8th and 11th graders (3465 girls and 3251 boys). The study population consisted of 39·83 % White/Other, 14·61 % African-American and 45·56 % Hispanic adolescents; with a mean age of 14·9 years, and 61·95 % at a healthy weight, 15·71 % having overweight and 22·34 % having obesity.
Results
Adolescents who did not use nutrition labels had 1·69 times greater odds of consuming ≥1 sugary beverages/d (P<0·05). Adolescents who used nutrition labels had 2·13 times greater odds of consuming ≥1 fruits and vegetables/d (P<0·05). Adolescents who used nutrition labels had significantly higher healthy eating scores than those who did not (P<0·001). For every 1-point increase in nutrition knowledge, adolescents had 1·22 greater odds of using nutrition labels.
Conclusions
Nutrition label use is associated with healthier dietary behaviours in adolescents. Intervention strategies for youth should include efforts to teach adolescents to use labels to make healthy food choices.
We compute the cohomology of the right generalised projective Stiefel manifolds. Following this, we discuss some easy applications of the computations to the ranks of complementary bundles and bounds on the span and immersibility.
Galton-Watson forests consisting of N roots (or trees) and n nonroot vertices are studied. The limit distributions of the number of leaves in such a forest are obtained. By a leaf we mean a vertex from which there are no arcs emanating.
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