3 types of PCA

Classical Principal Component Analysis

Finds a low dimensional linear subspace that maximises variance (or equivalently minimises reconstruction error). A purely algebraic operation. No notion of noise. No uncertainty in parameters

Probablistic Principal Component Analysis

Latent Factors

Reframes PCA a maximum likelihood under a probabilistic latent factor model. Still no priors. "Latent" here implies an unobserved \ hidden variables $\mathbf{z} \in \mathbb{R}^k, k≪d$ that cannot be measured directly. Instead the observed data is generated by said hidden variables.

Observations

Each observation is a vector: $\mathbf{x} \in \mathbb{R}^d$ and $n$ observations produce a matrix: $\mathbf{X} \in \mathbb{R}^{n \times d}$

Recovery

Recovery is the process of finding the most likely values of principal components given an observation using $\mathbf{z} = \mathbf{W}^T(\mathbf{x} - \boldsymbol{\mu})$. The maxiumum likelihood solution for $W$ recovers the classical PCA directions exactly. Generative model: $\mathbf{x} = \mathbf{W}\mathbf{z} + \boldsymbol{\mu} + \boldsymbol{\epsilon}$ Mean: $\boldsymbol{\mu} \in \mathbb{R}^d$ — mean of the observed data

Bayesian Principal Component Analaysis

Bayesian PCA is an extension of Probablistic PCA and therefore assumes that latent variables are normally distributed.

a word on the use of copulas by insurers

insurers can go down 2 routes: * independence assumption * or copula for joint behaviour

separating dependence structure from marginal behaviour
copula captures tail dependence that independence assumption misses

ICA VERSUS PCA

Principal Component Analysis

PCA finds components that are orthogonal and ordered by variance

Independent Component Analysis

ICA finds components that are statistically independent - a much stronger condition.

?? independence means that all higher moments (not just variance agree) ??

?? ICA exploits non-Gaussianity to identify components that are truly causally/generatively separate. ??

Gaussian Case

for the Gaussian case PCA and ICA are indistinguishable ... ICA only adds value when underlying sources are non Gaussian

advantages of PCA

• interpretability - intuitive mapping of components
• stability - PCA loadings are stable and well-understood; ICA solutions can be sensitive to initialisation and sample period
• regulatory acceptance

an argument for ICA

• data in non gaussian
• we are going to model as non-Gaussian
• we want components that are genuinely independent for scenario generation

ICA for yield curves

apparently PCAs orthogonal components are approximately independent

ICA gives rotated components that are not as economically interpretable as level, slope, curvature.

ICA has been explored academically but not implemented in practice

Methods used by Insurers to stress the yield curve

• standard formula
• stochastic models
• mean reversion models
• Vasicek
• Hull-White model (extension of Vasicek - better at initial fit)
• multi factor models
  • Principal Component Analysis
  • (actually a statistical factor model and not a structural factor model)
  • Heath Jarrow Morton

mean reversion models

definition

Actual use cases of PCA by Life Insurers

• duration matching
• hedging
• capital models

Regulatory Framework

questions

• where does Nelson-Seigel-Svensson fit in ?
• to what extent does regulatory framework influence our choice of model and discourage a novel approach

to-do

• verifying this information

Considering New approaches

Multiple Structural Models

I understand that it is possible to use more than one structural model and am therefore exploring the possibility of multiple likelihoods. I am not familiary with the matehmatics on this yet. However even with this approach an issue still remains insofar that I am trying to model principal components and as mentioned these dont have intuitive meaning as they existing in a transformed vector space.

The original goal was to produce a posterior distribution for the principal components. From this values could be drawn when performing simulations, a reverse transformation of these values into yield curve movements, which then generates a future yield curves. If I stick with this approach then perhaps I am still left with the problem of "no intuitive meaning" in trying to set priors. But perhaps the flexibillity of multiple likelihoods may help.

Transforming intuitive statements into multiple priors

An alternative may be to continue to think about deriving priors from intutiive statements by some transformation process... but I dont want this to be too complex, and I would like it to be able to accomodate multiple types expert judgements and so a one size fits all solution presents a challenge.