Gaussian Processes Sample Clauses

Gaussian Processes. (2018). xxxx://xxxxxx -xxxxx.xxx/xxxxxx/xxxxxxx/xxxxxxxx_xxxxxxx.xxxx. (Accessed: 10-06- 2019)

Gaussian Processes. ‌ Gaussian Processes (GP) has been a useful tool for scientists to perform non-parametric function inference for many years. Compared to the parametric approaches such as the least-squares fitting, it is advantageous in more flexible functional form and no empirical constraints on the amount of parameters [77]. Let us start by expanding function y = f (x) with respect to a basis set {Rnh} which are defined as the following: Rnh ≡ φh(xn) (4.4) Σ Rnh therefore indicates the h-th basis function centred on the variable xn, Accordingly, the function y is written as : yn ≡ Rnhωh (4.5) where {ωh} corresponds to the weight parameters for each of the basis function. The prior distribution of w = {ωh} is taken to be Gaussian type with zero mean and variance of σω: ω P (w) = Normal(w, σ2 · I) (4.6) Since y is the linear combination of w, its covariance matrix is : ω Q = yyT = RωωT RT = R ωωT RT = σ2 RRT (4.7) where R = {Rnh} indicates the basis set. The prior distribution of function y can therefore be expressed using the covariance matrix Q: P (y) = Normal(y, 0, Q) (4.8) error Introducing C as the noise-included covariance matrix: C = Q + I · σ2 , we obtain the following probabilities for tN and tN+1: N N Prior probability : P (tN ) ∝ exp − 1 tN C—1tT N +1  Joint probability : P (tN+1, tN ) ∝ exp − 1 [tN tN+1]C—1 [tN tN+1]T } In the above probabilities, CN is the covariance matrix built for the data set tN while CN+1 also includes the covariance with (N + 1)th measurement: tN+1.