Updated jupter notebooks

Author: Abhijnan03

books/.ipynb_checkpoints/krg-checkpoint.ipynb

@@ -0,0 +1,27 @@
+# Basic Surrogate Models
+
+This section provides implementation for concepts related to basic surrogate models. As discussed in the lecture, one of the ways to create a model of $y(x)$ is by considering a linearly weighted combination of basis functions. Mathematically, this can be represented as:
+
+$$
+    y(x) \sim \hat{y}(x) = \mathbf{w}^T\pmb{\psi} = \sum w_i \psi_i(x)
+$$
+
+where $\mathbf{\pmb{\psi}}$ is the vector of basis functions and $\mathbf{w}$ is the vector of weights. The model is linear in terms of weights $w_i$ but basis function $\psi_i(x)$ can be non-linear. For example, the basis functions vector $\pmb{\psi}$ can be $[1 \text{  } x \text{  } x^2 \text{  } \sin(x) \text{  } e^{x}]^T$. The weights $w_i$ are determined by minimizing the sum of squared differences between the predictions and actual values. The weights obtained after minimization are given by (refer lecture notes for derivation):
+
+$$
+    \mathbf{w} = (\Psi^T\Psi)^{-1}\Psi^T\mathbf{y} = \Psi^{\dagger}\mathbf{y}
+$$
+
+where $\mathbf{y}$ is the vector of target values and $\Psi^{\dagger}$ is the (Moore-Penrose) pseudo-inverse of $\Psi$ matrix. The pseudo-inverse will be regular inverse if $\Psi$ is invertible. The performance of the model can be evaluated using the mean squared error (MSE) which is given by:
+
+$$
+    \text{RMSE} = \sqrt{ \frac{1}{N}\sum_{i=1}^{N} (y_i - \hat{y}_i)^2 }
+$$
+
+where $N$ is the number of data points, $y_i$ and $\hat{y}_i$ are the true and predicted values respectively.
+
+In this section, following models are covered:
+
+1. Linear model
+2. Polynomial model
+3. Radial basis function model

books/basic_sm/.ipynb_checkpoints/intro-checkpoint.md

@@ -1,6 +1,6 @@
 # Basic Surrogate Models
 
-This section provides implementation for the concepts covered in the class for basic surrogate models. As discussed in the lecture, one of the ways to create a model of $y(x)$ is by considering a linearly weighted combination of basis functions. Mathematically, this can be represented as:
+This section provides implementation for concepts related to basic surrogate models. As discussed in the lecture, one of the ways to create a model of $y(x)$ is by considering a linearly weighted combination of basis functions. Mathematically, this can be represented as:
 
 $$
     y(x) \sim \hat{y}(x) = \mathbf{w}^T\pmb{\psi} = \sum w_i \psi_i(x)

books/basic_sm/.ipynb_checkpoints/ls_regression-checkpoint.ipynb

@@ -6,7 +6,7 @@
    "source": [
     "# Polynomial Model\n",
     "\n",
-    "This section supports material covered in the class for polynomial models. Below block of code imports required packages."
+    "This section provides implementation for concepts related to polynomial models. Below block of code imports required packages."
    ]
   },
   {
@@ -650,7 +650,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "sm",
+   "display_name": "Python 3 (ipykernel)",
    "language": "python",
    "name": "python3"
   },
@@ -664,9 +664,9 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.12.0"
+   "version": "3.9.19"
   }
  },
  "nbformat": 4,
- "nbformat_minor": 2
+ "nbformat_minor": 4
 }

books/basic_sm/.ipynb_checkpoints/poly-checkpoint.ipynb

@@ -6,7 +6,7 @@
    "source": [
     "# Radial Basis Function Model\n",
     "\n",
-    "This section supports material covered in the class for radial basis function (RBF) model. In these models, a specific type of basis functions is used which computes similarity between inputs and a set of reference points in the design space. Let $\\mathbf{x}$ be the input vector, $\\mathbf{c}_i$ be the $i^{th}$ reference point vector, and $r_i$ be the euclidean distance between $\\mathbf{x}$ and $\\mathbf{c}_i$ i.e. $|| \\mathbf{x} - \\mathbf{c}_i ||_2$. Then, the basis function vector will be:\n",
+    "This section provides implementation for concepts related to radial basis function (RBF) model. In these models, a specific type of basis functions is used which computes similarity between inputs and a set of reference points in the design space. Let $\\mathbf{x}$ be the input vector, $\\mathbf{c}_i$ be the $i^{th}$ reference point vector, and $r_i$ be the euclidean distance between $\\mathbf{x}$ and $\\mathbf{c}_i$ i.e. $|| \\mathbf{x} - \\mathbf{c}_i ||_2$. Then, the basis function vector will be:\n",
     "\n",
     "$$\n",
     "    \\pmb{\\psi} = [ \\psi_1(r_1) \\quad  \\cdots \\quad \\psi_i(r_i) \\quad \\cdots \\quad \\psi_n(r_n)]^T\n",
@@ -510,7 +510,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "sm",
+   "display_name": "Python 3 (ipykernel)",
    "language": "python",
    "name": "python3"
   },
@@ -524,9 +524,9 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.12.0"
+   "version": "3.9.19"
   }
  },
  "nbformat": 4,
- "nbformat_minor": 2
+ "nbformat_minor": 4
 }