ST211 Lecture 2.pdf

# ST211 Lecture 2.pdf - ST211 Applied Regression Simple...

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ST211 Applied Regression

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We could say a lot about what is meant by a model, and the modelling process. We condense the discussion by saying that many natural world phenomena can be represented by a mathematical equation, the model: • E = mc 2 (Relativity Law) • v 2 = u 2 +2as (Newton’s Laws of Motion) • Production = K* Labour α (Cobb-Douglas Relation) • H = Acoswt + Bsinwt (Height of tide in a harbour) • D = α + βA + γA 2 (Drying time of paint as a function of amount of additive used) • V = β 0 1 F + β 2 B + β 3 G + β 4 A + … (Value of a property in terms of floorspace F, number of bathrooms B, garden size G, age A and more) etc. Simple Linear Regression (SLR)
Simple Linear Regression (SLR) In its basic form, simple linear regression involves finding – and then analysing – a model that takes the form y = α + βx. We later include an error term to account for Imprecision of the estimates we find. Such models are simple because there is only a single predictor x, in contrast with multiple regression with several predictors; and linear because y is a linear function of x and not x 3 , log x or whatever. We shall look briefly at some important features of SLR. However, the main part of the description will be in the context of multiple regression, which is really an extension of SLR, as we shall see.

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Simple Linear Regression When we want to carry out SLR, we should routinely produce a scatterplot. This will provide a quick check on linearity as well as drawing out any unusual features of the data, such as outliers. Linear models do occur in practice: fuel consumption of aircraft with distance travelled weight by height of a group of animals value of a property from area of floor space sales of scarves by temperature etc. But as we shall see later, an improved model may generally be achieved by including other variables, such as garden size and age of a property as well as floor space.
Simple Linear Regression We assume that we are given pairs of data (x 1 , y 1 ), (x 2 , y 2 ), … (x n , y n ). We seek the ‘line of best fit’ for these data, that is, the line y = α + βx for which the distance from points to line is as small as possible. A natural criterion is to use Here LSE, stands for Least Squares Estimate, y obs is an observed value of y, while y mod is the corresponding modelled value.

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