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previous page next page A couple of quotes BACK FORWARD 1 previous page next page Preface statistics (descriptive or explora...

previous page next page A couple of quotes BACK FORWARD 1 previous page next page Preface statistics (descriptive or exploratory vs inferential) from a user (engineer or scientist) point of view – role of statistics in engineering/mechanics/science – ‘big’ data vs ‘small’ data: data analytics ∗ machine (statistical) learning (AI) vs traditional inference aims of data analysis – prediction vs understanding – correlation vs causation – statistical significance vs practical significance; role of theory – data analysis for design – Occam’s razor: simplest (parsimonious) model for the purpose survey of statistical techniques and tests – judicious use and interpretation of results, understanding limitations – underlying/recurring concepts prerequisites? undergraduate statistics, calculus, matrix algebra, programming BACK FORWARD 2 previous page next page Computational statistics software environments such as R facilitates traditional (and non- traditional) analyses – understanding physical relationships as manifested in data, data visualization computer simulations – testing assumptions of statistical models statistical techniques made practical by computers – maximum likelihood, bootstrapping, Bayesian approaches – machine (statistical) learning (artificial intelligence) – questioning of traditional approaches BACK FORWARD 3 previous page next page Preliminaries problem statement(s): what can be learned (or inferred) from a set of (sampled) data? what is the uncertainty in quantitative estimates? Is there sufficiently strong empirical support for a given hypothesis? statistics as means of making sense of random phenomena: what is randomness? – unpredictability – modeling physical phenomena and relationships: deterministic and random elements y = f (x; β) +  where y is a scalar (single) dependent variable (also termed response variable), x is a vector of (multiple) independent variables, β a vector of model parameters, and  is a random variable ∗ f (x; β) is ‘deterministic’ model based on physics/mechanics (maybe),  - statistical model; x usually considered perfectly known, β ‘true’ parameters to be estimated (fitted) to data ∗ statistical model often makes a distribution assumption (parametric model), but could be distribution-free (non-parametric) BACK FORWARD 4 previous page next page Sample and Population population: all possible realizations of an experiment – statistical homogeneity (or stationarity)? – population parameters (characteristics): not random (at least conceptually), but usually unknown or even unknowable what does given data represent? – sample: subset of (usually much smaller than but possibly identical to) a population ∗ representative sample: relevant statistics of sample statistically similar to population characteristics ∗ (simple) random sample: each point in population has an equal chance of being sampled ∗ implications of samples that are not necessarily representative nor random – bias – which population is the ‘target’ of sample data? ∗ effects of scale (or possibly other external effects) ∗ dimensionless (or normalized) characteristics to minimize effects of scale and to broaden population BACK FORWARD 5 previous page next page Descriptive and Inferential Statistics Descriptive statistics (also exploratory data analysis) – focus on sample, no (immediate) concern for relationship to any population – appropriate when understanding of data is low – possible preliminary to inferential statistics inferential statistics – going beyond the sample: relationship(s) between sample statistics and population parameters – some understanding of data available; meaningful formulation and testing of hypotheses ∗ plausible physical and statistical model assumptions can be made about data – random aspects =⇒ relationships between sample and population characteristics must be framed in probabilistic terms BACK FORWARD 6 previous page next page Review: Basic probability concepts and terminology I random variable, X, that takes on a value, x, (or between values, x − (dx/2) and x + (dx/2)) in a random manner – different types: numerical (discrete, continuous), categorical (ordered and non-ordered) – function of random variables must be random variable =⇒ sample statistic (a function of random variables) is a random variable probability statements about (continuous) X:   dx dx P x− 1, central moments about µx: E[(X − µx) ] = (X − µx)m f (x)dx – variance, σx2 = E[(X − µx)2] (σx = standard deviation is a possibly dimensional scale or range parameter, σx/µx = coefficient of variation) – normalized third central moment, gSk = E[(X − µx)3]/σx3 ; for distributions symmetric about µx like the normal gSk = 0 – normalized fourth central moment, gKu = E[(X − µx)4]/σx4 ; for normal distribution gKu = 3, so excess kurtosis = gKu − 3 > 0 implies greater (than normal) importance of distribution tails (normalized) higher-order (> 2) moments give more information about and are more influenced by distribution tails, but sample estimates are correspondingly more uncertain standard distributions defined by limited number (usually