appears to be an uncanny fit (orange) to 400 measured values (blue) using 16 sinusoidal frequencies! this time it's grant work, not contract work, so i will be able to explain this in more detail eventually... for now, it's just a work in progress that looks cool :)

#dataScience #dataAnalysis #spectralAnalysis #leastSquares #OLS #sinusoidal #trigonometric #regression #modeling #statistical

`Introduction to the Cox model`

https://www.youtube.com/watch?v=CzHt22YL_-w

#coxModel #proportionalHazardModel #davidCox #sirDavidCox #survivalAnalysis #failureAnalysis #failureTimeAnalysis #forceOfMortality #reliabilityEngineering #kaplanMeier #failureDistribution #failureTimeDistribution #failureRate #hazardFunction #hazardRate #failureRate #nonParametric #model #modeling #statistics #stats #mortality #function #mathematics #math #proportionality #regression #regressionModel #leastSquares

Last lecture 🔖 of this semester is done! Interestingly the last topic was linear regression 📈 and correlation. A fascinating topic with lots to say about #Leastsquares and scientific #fitting of data in general. But with little time ⌛ and little #mathematics it's hard to outline how useful this approach can be.

Looking forward to the mini projects results the students have to hand in soon. Hopefully they learned a little about #statistics 😅

The #VectorAutoRegression stuff I've been doing can be summarized as
$$ y_t = \sum_{i=1}^p A_i y_{t - i} $$
where each $y_t$ is a $D$-vector and each $A_i$ is a $D \times D$ matrix.

Given an input series of $y$ values, the $A_i$ can be calculated by #LeastSquares minimization:

$$
Y = [ y_p ; y_1; \ldots ; y_{T - 1} ] \\
X = [ y_{p-1}, y_{p-2}, \ldots, y_0 ; y_p, y_{p-1}, \ldots, y_1 ; \ldots ; y_{T-2}, y_{T-3}, \ldots, y_{T-1 - p} ] \\
A = \left(X^T X\right)^{-1} X^T Y
$$
where the matrices have these initial dimensions:
Y : (T - p) × D
X : (T - p) × (p × D)
A : (p × D) × D
then reshape $A$ to $p × (D × D)$

In my case all values are complex (and ^T is conjugate transpose) because each $y$ is an FFT of a block of input data - I'm using FFT size $256$ (making $D = 129 = 256/2+1$ unique bins for real input) overlapped 4x.