Math 156

Perturbation

Measure

  • \(\mu:\{A\subset \Omega\}\to [-\infty,\infty]^?\), i.e. maybe some subsets are not measurable.
  • \(\mu(\emptyset)=0\)
  • If \(A\) has measurement then \(\mu(\Omega-A\)\) has measurement.
  • If \(A_1, A_2,\ldots\) disjoint then \(\mu(A_1\sqcup A_2\sqcup \cdot )=\mu(A_1)+\mu(A_2)+\cdots\)

\[F(t)=\int g(x,t) d\mu\]

  • Let \(x\) be a variable that spans a data set, e.g. measurements of individuals in a study.
  • Let \(t\) be a parameter in the measurements.
  • We aggregate measurements by adding them up.
  • How to add?  A weighted average? Some command in Java called Sum?  .... just use \(\int-d\mu\) symbol as stand in for a sum

\[F(t)=\int g(x,t) d\mu=\sum_{i=1}^n |x_i-t|\frac{1}{n}\]

Discrete measurements: data \[x_1,\ldots,x_n\]

What does \(F(t)\) measure?

\[F(t)=\int g(x,t) d\mu=\sum_{i=1}^n |x_i-t|\frac{1}{n}\]

Discrete measurements: data \[x_1,\ldots,x_n\]

What does \(F(t)\) measure?

On average how much does \(x_i\) deviate from \(t\)

\[F(t)=\int g(x,t) d\mu=\sum_{i=1}^n (x_i-t)^2\frac{1}{n}\]

Tuning our feature \(F\) using \(g(x,t)=(x-t)^2\)

What does \(F(t)\) measure?

\[F(t)=\int g(x,t) d\mu=\sum_{i=1}^n (x_i-t)^2\frac{1}{n}\]

What does \(F(t)\) measure? Not obvious, lets look at the rate of change in t

\[\frac{\Delta F}{\Delta t}=\frac{F(t_1)-F(t_0)}{t_1-t_0}=....\]

\[F(t)=\int g(x,t) d\mu=\sum_{i=1}^n (x_i-t)^2\frac{1}{n}\]

What does \(F(t)\) measure? Not obvious, lets look at the rate of change in t

\[\frac{\Delta F}{\Delta t}=\frac{F(t_1)-F(t_0)}{t_1-t_0}\approx 2 \sum(x_i-t)\frac{1}{n}\]

This we already understood...so the rate of change is measuring....?

\[F(t)=\int g(x,t) d\mu=\sum_{i=1}^n (x_i-t)^2\frac{1}{n}\]

Rectangular measurements: data \[x_1,\ldots,x_n\]

What does \(F(t)\) measure?