Statistics 359a
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Statistics 359a. Regression Analysis. Necessary Background Knowledge - Statistics. expectations of sums variances of sums distributions of sums of normal random variables t distribution – assumptions and use calculation of confidence intervals simple tests of hypotheses and p-values.

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Statistics 359a

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Statistics 359a

Statistics 359a

Regression Analysis


Necessary background knowledge statistics

Necessary Background Knowledge - Statistics

  • expectations of sums

  • variances of sums

  • distributions of sums of normal random variables

  • t distribution – assumptions and use

  • calculation of confidence intervals

  • simple tests of hypotheses and p-values


Necessary background knowledge linear algebra

Necessary Background Knowledge – Linear Algebra

  • multiplication of conformable matrices

  • transpose of a matrix

  • determinant of a square matrix

  • inverse of a square matrix

  • eigenvalues of a square matrix

  • quadratic forms


Origin of least squares

Origin of Least Squares

Introduction of the metric system and the length of a meter

  • 1790 – French National Assembly commissions the French Academy of Sciences to design a simple decimal-based system of weights and measures

  • 1791 – French Academy defines the meter to be 10-7 or one ten-millionth of the length of the meridian through Paris from the north pole to the equator.


Adrien marie legendre

Adrien-Marie Legendre

  • Legendre on the French commission in 1792 to determine the length of the meridian quadrant

  • measurements of latitude made in 1795

  • complex calculations made from the measurements in 1799

  • Legendre proposes the method of least squares in 1805 to determine the length of a meter


Statistics 359a

Data

  • old French units of measurement: 1 module = 2 toises

  • old French to imperial English: 1 toise = 6.395 feet

  • metric to imperial: 1 meter = 3.2808 feet


From spherical geometry

From Spherical Geometry


Including measurement errors the data and model reduce to

Including measurement errors, the data and model reduce to:


Solution is

Solution is:

D = 28497.78 modules

90D = 2564800.2 modules = length of the meridian quadrant

Therefore

1 meter = 0.256480 modules

= 0.512960 toises

= 3.280 feet

modern meter = 3.2808 feet


Origin of the term regression

Origin of the Term “Regression”

  • Francis Galton, 1886, ‘Regression towards mediocrity in hereditary stature.’ Journal of the Anthropological Institute, 15: 246 – 263

  • See JSTOR under UWO library databases


Data on heights of children and parents

Data on Heights of Children and Parents


Regression line

‘Regression Line’


Theoretical basis

Theoretical Basis

For X and Y bivariate normal with equal means variances

For  > 0

E(Y |X ) < x for x >  and

E(Y |X ) > x for x < 


Example in data analysis through regression

Example in Data Analysis Through Regression

  • Relationship between the price of a violin bow and its attributes such as age, shape and ornamentation on the bow


Price and date of sale

Price and Date of Sale

  • 1995 seems to be a more expensive year

  • Is the effect confounded with some other attribute common to 1995?


Price and year of manufacture

Price and Year of Manufacture

  • Is there anything special about 1920?

  • Is there a quadratic trend in the data?


Price and weight of the bow

Price and Weight of the Bow

  • Is there any trend with respect to the weight?


Octagonal vs round bows

Octagonal vs. Round Bows

  • No apparent trend


The gold standard

The Gold Standard?

  • The presence of gold on a bow generally makes it more expensive


Tortoise shell frogs

Tortoise Shell Frogs

  • Some evidence of added expense for tortoise shell


Price and pearl accessories

Price and Pearl Accessories

  • No apparent effect


Prediction

Can we use the model built with the current data to predict the future price of a bow

Example: some 1999 data from auctions

1920 bow, 60.5 g., round with gold and pearl accessories - $4098

1933 bow, 61 g., octagonal with pearl accessories only - $2421

Prediction


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