Publications by Jake
smoothers
Smoothers Jake 02/07/2021 Scatterplot Smoothing Scatterplot smoothing is a non-parametric method with a single predictor. The scatterplot points are treated only as points on a plane. without regard to an underlying probabalistic model Assumes \(x_i's\) are distinct and ordered \[ y_i = f(x_i) + \epsilon_i\] Running Mean Running mean smoot...
17619 sym R (4054 sym/14 pcs) 9 img
linear reg
Linear Regression Jake 31/05/2021 Sum of Squares Notes: \[ S_{xx} = \sum^n_{i=1}(x_i-\bar{x})^2,\quad s_{x}^2 = \frac{S_{xx}}{n-1}\] \[ S_{yy} = \sum^n_{i=1}(y_i-\bar{y})^2,\quad s_{y}^2 = \frac{S_{yy}}{n-1}\] \[ S_{xy} = \sum^n_{i=1}(x_i-\bar{x})(y_i-\bar{y}),\quad s_{xy} = \frac{S_{xy}}{n-1}\] Simple Linear Regression The simple linear regre...
18320 sym R (4686 sym/45 pcs) 15 img
General Linear
Generalised Linear Models Jake 26/06/2021 Generalised Linear Models The goal of generalised linear models is to model a transformation of the mean response linearly. This allows for a lot of the guassian model assumptions to be relaxed and opens up the possibility of the use of other distributions. Consider the following idea, where \(g(\mu_i...
11864 sym R (8754 sym/26 pcs) 1 img
NN
Neural Network Notes Jake 29/11/2021 Gradient Descent Gradient descent is an optimization technique that is used to minimize a function, as long as the derivative can be found. It is often used when we can’t find the minimum of a function analytically. The algorithm Initialise the algorithm with the following parameters \(\theta\), the par...
8044 sym 3 img
CAPM and Factor Models
CAPM And Factor Models Jake Warby 17/11/2021 CAPM The CAPM model considers the economic concept of equilibrium Focused on the analysis of singular assets with the mean-variance approach The assumptions for the model is as follows: Assumptions Assumptions Cont Individual Optimisation Individuals are faced with the problem of maximising uti...
12457 sym 4 img
Discrete Derivatives
Discrete Derivatives Jake 06/11/2021 Derivatives Derivatives are securities where the future payoff relies on the underlying security. Examples include forwards, options, futures, swaps etc. Valuing Deriviatives A deriviative payoff can be seen as a function of the underlying asset \[ f(S_T)\] An example of a derivative is the forward con...
10731 sym 5 img
MPT
Modern Portfolio Theory Jake 21/10/2021 Utility Theory Utility is used to measure an investor’s preferences. For deterministic outcomes, the following means that asset x is preferred over asset y: \[ U(x) > U(y)\] Often outcomes are uncertain, being phrased as gambles: \[ G(A,B;\alpha) = \begin{cases}A,\text{ with prob }\alpha\\ B,\text{ ...
10924 sym 1 img
Density Estimation
Density Estimation Jake 19/08/2021 Estimation of Unknown Density Parametric density estimations are one approach to estimation of an unknown density, however they often miss significant structures within data For example normal is unimodal however the data may be multimodal Histograms can be used as a basic density estimation technique Bin...
7176 sym R (735 sym/5 pcs) 5 img
Bayesian Models
Bayesian Models Jake 19/08/2021 Bayesian Inference In frequentist theory, \(\theta\) is treated as a constant and the data is treated as random. \[ p(x|\theta) = X|\theta\sim Bin(n,\theta) \] In Bayesian theory, \(\theta\) is treated as a random quantity, inferences are made in terms of probability statements. \[ p(\underbrace{\theta}_{\text...
6879 sym R (1011 sym/10 pcs) 10 img
Simulation
Simulation Jake 20/08/2021 Simulation Methods From a uniform random variable \(U\sim Uni(0,1)\) we can generate a RV from any arbitrary distribution or approximate integrals. The uniform random variable, with density defined as follows, is our source of randomness for further simulations. \[ f(u) = \begin{cases}1, \quad\text{if }u\in[0,1]\\0,\...
10397 sym R (1887 sym/26 pcs) 2 img