statistics 359a
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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
• 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 SquaresIntroduction 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• 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
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
earth theofy ellipticit the torelated is modules in
arc an of degree one of length )28500/(1 D
(90D)quadrant meridian theof length the torelated is length arc
)cos()sin(2850028500
CC
C
S
LLLLSSLL
Including measurement errors, the data and model reduce to:
)014.0()765.4(000279.0 )277.0()914.2(001529.0 )324.0()048.0(002625.0 )027.0()720.2(000475.0 )590.0()912.4(003398.0
5
4
3
2
1
CCCCC
Solution is:
D = 28497.78 modules90D = 2564800.2 modules = length of the
meridian quadrantTherefore
1 meter = 0.256480 modules = 0.512960 toises = 3.280 feetmodern meter = 3.2808 feet
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
Theoretical Basis
For X and Y bivariate normal with equal means variances
For > 0E(Y |X ) < x for x > and E(Y |X ) > x for x <
)()|( xxXYE
))(1()|( xxxXYE
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
Violin Bow Example
The following data on violin bows made by W.E. Hill and Sons of London, England are taken from the internet site www.maestronet.com/pricehist.html. The data show the prices of the bows sold at auction at Sotheby’s auction house for the years 1994-97. Also given are data on various factors that may affect the price of the bow. These include: the year of the sale (in case of price inflation or deflation); the year of manufacture (or age – are antique bows more or less valuable?); weight of the bow in grams (do buyers like heavier or lighter bows?); the shape of the bow (is there an aesthetic effect to the price?); presence or absence of ornamental gold; presence or absence of ornamental pearl; and whether the bow has a tortoiseshell frog or an ebony frog. Only the bows for which the approximate year of manufacture has been given are included in the data set. Prices from other auction houses and for other bow makers, as well as violins, are available at the same site, but only Sotheby’s gives the year of manufacture. A Minitab file of the data is at O:\359\bows.mtb.
Price in U.S.
Dollars Year of
Sale
Year the Bow was
Made Weight in
Grams
Shape O=octagonal
R=round Gold
Accessories
Tortoise-shell Frog
Pearl Accessories
1874 1997 1957 59.0 O N N N 2436 1997 1935 62.0 R N N N 7498 1997 1920 62.0 R Y Y N 1142 1996 1945 59.5 O N N Y 1935 1996 1890 57.5 R N N N 1759 1996 1900 56.0 O N N N 5278 1996 1950 57.0 O Y Y Y 4905 1995 1920 58.0 R Y N N 7994 1995 1920 60.0 O Y Y Y 2543 1995 1926 62.5 R N N Y 1769 1994 1935 61.0 R N N N 1592 1994 1960 61.0 R N N Y 3716 1994 1935 55.0 O Y Y Y 2477 1994 1925 59.0 R N N Y 2654 1994 1930 58.0 R N N N 3362 1994 1935 58.0 R N Y Y
Price and Date of Sale• 1995 seems to be a more expensive year• Is the effect confounded with some other attribute
common to 1995?
1997199619951994
8000
7000
6000
5000
4000
3000
2000
1000
Year Sold
Price
Violin Bows - Price and Sale Date
Price and Year of Manufacture
• Is there anything special about 1920?• Is there a quadratic trend in the data?
1890 1900 1910 1920 1930 1940 1950 1960
1000
2000
3000
4000
5000
6000
7000
8000
Year Made
Price
Violin Bows - Price and Year of Manufacture
Price and Weight of the Bow
• Is there any trend with respect to the weight?
636261605958575655
8000
7000
6000
5000
4000
3000
2000
1000
Weight
Pric
e
Violin Bows - Price and Weight in Grams
Octagonal vs. Round Bows
• No apparent trend
80007000600050004000300020001000
1.0
0.5
0.0
Price
Shap
e
Violin Bows - Price and Shape1 = round, 0 = octagonal
The Gold Standard?
• The presence of gold on a bow generally makes it more expensive
80007000600050004000300020001000
1.0
0.5
0.0
Price
Gol
d
Violin Bows - Price and Gold Accessories1 = present, 0 = absent
Tortoise Shell Frogs
• Some evidence of added expense for tortoise shell
80007000600050004000300020001000
1.0
0.5
0.0
Price
Frog
Violin Bows - Price and Tortoise Shell Frogs1 = present, 0 = absent
Price and Pearl Accessories
• No apparent effect
80007000600050004000300020001000
1.0
0.5
0.0
Price
Pea
rl
Violin Bows - Price and Pearl Accessories1 = present, 0 = absent
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