Covariance between x and f(x)?

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In summary, the conversation discusses determining propagated error for a function involving two variables, x and t. The first attempt was done with uncorrelated x and t, but now the given x is a function of t. The equation for x(t) is x(t) = a √t, where a is a constant. The solution involves finding the uncertainty in g(t), defined as f(x(t),t), using g'(t) and the chain rule. This eliminates the need for calculating the covariance between t and x(t).
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hb1547
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Homework Statement


As part of an assignment, I have to determine propagated error of some function:
[itex]f(x,t)[/itex]
I did it first with x & t being completely uncorrelated, but now I'm given x as a function of t, [itex]x(t)[/itex], and have to do the same.

Homework Equations


I know the linear approximation for finding the uncertainty [itex]\sigma_f[/itex], all I need is the covariance term.
The equation x(t) itself is simply [itex]x(t) = a \sqrt{t}[/itex], where a is just a constant.

The Attempt at a Solution


I seem to be having a hell of a time Googling to find a way to calculate the covariance between t and x(t).
 
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  • #2
Hi hb1547! :smile:

I do not believe you need the covariance.
Let's define g(t)=f(x(t),t).
Then the uncertainty in g(t) is g'(t)σt.

Do you know how to calculate g'(t)?
It consists of a couple of partial derivatives and the chain rule...
 

1. What is covariance between x and f(x)?

Covariance between x and f(x) is a measure of how two variables (x and f(x)) change together. It indicates the direction and strength of the relationship between the two variables. A positive covariance indicates that the variables tend to move in the same direction, while a negative covariance indicates that they tend to move in opposite directions.

2. How is covariance calculated?

Covariance is calculated by taking the sum of the products of the differences between each data point and the mean of each variable. This sum is then divided by the total number of data points. The formula for covariance is: cov(x,f(x)) = Σ(x - μx)(f(x) - μf(x)) / n

3. What does a high covariance between x and f(x) mean?

A high covariance between x and f(x) means that there is a strong relationship between the two variables. This could indicate that as one variable increases, the other also tends to increase (positive covariance), or as one variable increases, the other tends to decrease (negative covariance). However, covariance alone does not tell us the strength or direction of the relationship, so it is important to also look at the correlation coefficient.

4. How is covariance different from correlation?

Covariance and correlation are both measures of the relationship between two variables. However, covariance measures the strength and direction of the relationship, while correlation also standardizes the data to a scale of -1 to 1, making it easier to interpret. Additionally, correlation is not affected by changes in the scale or units of the variables, while covariance is influenced by the units of the variables.

5. What can we infer from a covariance of zero between x and f(x)?

A covariance of zero between x and f(x) means that there is no linear relationship between the two variables. This does not necessarily mean that there is no relationship at all, as there could be a non-linear relationship between the variables. To fully understand the relationship between x and f(x), further analysis using techniques such as correlation or regression may be necessary.

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