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Summary of introduced statistical terms and concepts

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Slides 6, 17 updated 2014-03-31

Summary of introduced statistical terms

and concepts

mean

Describes/measures

average conditions or the

center of the sample points

Variance

&

standard

deviation

Describes/measures

the spread of the sample points;

deviations from the

center of the sample points

covariance

&

correlation

Describes/measures

co-dependence of variations

in samples of two random

variables

Summary of introduced statistical terms

and concepts

Calculated mean values:

unbound, any real number

(check your values:

it must be within the minimum

and maximum of the sample data)

mean

Variance

&

standard

deviation

Variance: values are > or = 0

Standard deviation: > or = 0

(check your values: standard deviation

should be less than the minimum-maximum

sample range |max(x)-min(x)|)

covariance

&

correlation

Covariance: any real number

Correlation: between -1 and +1

(check your values: correlation should

never exceed the range from -1 to 1]

Linear function: y= bx +a

The value of y depends on the value of x

Δy = b*Δx

Δx

Note: I corrected the notation of the equation, please check your notes b is the slope,

a is the constant the intercept value. R-script class14.R was updated (2014-03-25 4:30pm)

Linear function y= bx +a

The value of y depends on the value of x

b= Δy/Δx =2

Δy = b*Δx

Δx

a=-1

Linear function y= bx +a

The value of y depends on the value of x

value of y

does not depend on x

a= Δy/Δx =2

y= 0x + a= a

Note: updated slide to define y with random error

?

How to estimate the best fitting line?

How to estimate the best fitting line?

Mathematically we formulate this as a minimization problem:

Minimize the distance of the data points from the linear regression line.

How to estimate the best fitting line?

The deviations from

the deterministic model

line are interpreted as random errors (following a Gaussian distribution)

Sum of Squared Errors (SSE)

How to estimate the best fitting line?

Sum of Squared Errors (SSE)

Intercept

Slope

How to estimate the best fitting line?

Sum of Squared Errors (SSE)

Sample mean of x and y

^

^

^

Note:

Many textbooks and statisticians would prefer to distinguish the estimated values

from the actual true (but unknown) parameter values using a different symbol.

Or they use Greek letters for the true values, and Latin letters for the estimates.

1

n

1

n

How to estimate the best fitting line?

Sum of Squared Errors (SSE)

COV(x,y)

b=

VAR(x)

1

n

1

n

How to estimate the best fitting line?

Sum of Squared Errors (SSE)

Slope of the regression line:

Correlation coefficient * standard deviation (y) / standard deviation (x)

How to estimate the best fitting line?

Estimated

Regression line

Linear relationship with errors: y= bx +a + ε

The value of y depends on the value of x plus a random error

Class exercises:

download script class14.R:

source(“class14.R”)

(1) change the linear parameters to have steeper, or more flat slopes.

(2) change the slope to negative (from top left to bottom right)

(3) observe how the correlation coefficient changes

(4) change the error variance and observe how it affects the correlation

and fitting of the line

(5) watch in case, where does the line intersect with the y-axis

(6) change the intercept parameter (intersection with the y-axis).

What is the effect on the correlation?

(7) find a way to change the sample size of the scatter points and repeat

your (1)-(6)

(8) set the slope parameter closer to 0 and eventually to 0 change the variance

of the errors. What happens to correlation?

Note: In R-scripts the variable names have a slightly different notation:

As you can see in class14.R we use ‘yobs’ for the variable y containing

the random error ‘e’.

The estimator for the slope ‘bfit’ is calculated by using for the correlation

coefficient ‘cor(x,yobs)’ and the standard deviation ‘sd(yobs)’ and ‘sd(x)’

The equation on slide 14

includes the error