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?
Copy of Measurements
By James Wilson
Copy of Measurements
- 7
