In the study of measure transformation in Gaussian space, there is a fundamental issue that I want to write down here, which is Fredholm determinant. Let me try to use half an hour to explain the intuition of this determinant in matrix form with finite dimension.
So suppose a matrix , our goal is to find out what definition is that makes sense for trace class operator(i.e.). We know that its trace , and assume the eigenvalues of are , we have
Iet’s take a look at these terms in the summation,
when , it’s 1,
when , it’s ,
when , it’s ,
which is the trace of operator on a linear space with basis where is is the wedge product form.
Similarly, we have the expression given in widipedia about “Fredholm determinant” that for a general trace-class operator
and this new operator is a linear operator on space formed by the basis
Finally, I want to mention that when , easy to check that indeed.
Today in class we talked about Two-way ANOVA, so my question is how to understand the essence of these two kinds of ANOVA? I’ll try to spend less than one hour to write down what I thought about it.
Let’s take a look at an example, we have treatments and groups and for each treatment and group , we have samples, so our two-way ANOVA model should would be
where is the mean of the th treatment and th group.
What if we want to use one-way ANOVA to model it? What would that be look like? Because there may be interaction between the treatment and group, so in my understanding, we should consider this model be a linear model with respect to three terms, i.e. treatment level , group level and interaction level , so the model should be , now we consider the least square solution for we have
By taking derivatives with respect to , we have
Then the solution of these equations is the two-way ANONA estimator.
So as we can see that if there is only one factor, there would not exist the interaction term , then we can just regard the original linear model be different group and analyse individually.
So in sum, so called “Two-way ANOVA” is mainly dealing with the interaction between two different “factors”. So question is if there are three different factors, can we follow the same idea do the “many-factors ANOVA”?