removed equation numbering

Author: Dekermanjian

examples/case_studies/ssm_hurricane_tracking.ipynb

@@ -22,13 +22,14 @@
     "In this case study we are going to forecast the paths of hurricanes by applying several State Space Models (SSM). We will begin with a simple two-dimensional constant acceleration tracking model, where we only have one parameter to estimate. Subsequently, we will progressively add complexity and parameters as we develop up our model. \n",
     "\n",
     "As a brief introduction to SSMs, the general idea is that we define our system using two equations.<br> \n",
-    "The state equation [1] and the observation equation [2]. \n",
+    "The state equation and the observation equation. \n",
+    "\n",
     "$$\n",
-    "x_{t+1} = A_{t}x_{t} + c_{t} + R_{t}\\epsilon_{t} \\quad (1)\n",
+    "x_{t+1} = A_{t}x_{t} + c_{t} + R_{t}\\epsilon_{t}\n",
     "$$ \n",
     "\n",
     "$$\n",
-    "y_{t} = Z_{t}x_{t} + d_{t} + \\eta_{t} \\quad (2)\n",
+    "y_{t} = Z_{t}x_{t} + d_{t} + \\eta_{t}\n",
     "$$\n",
     "\n",
     "The process/state covariance is given by $\\epsilon_{t} \\sim N(0, Q_{t})$ where $Q_{t}$ is the process/state innovations and the observation/measurement covariance is given by $\\eta_{t} \\sim N(0, H_{t})$ where $H_{t}$ describes the uncertainty in the measurement device or measurement procedure. \n",
@@ -63,11 +64,11 @@
     "The following equations define the process:\n",
     "|Description|Equation|\n",
     "| --- | --- |\n",
-    "|Predict the next state vector| $\\hat{x}_{t+1\\|t} = A_{t}\\hat{x}_{t\\|t} \\quad (3)$ |\n",
-    "|Predict the next state/process covariance| $P_{t+1\\|t} = A_{t}P_{t+1\\|t}A_{t}^{T} + Q \\quad (4)$ |\n",
-    "|Compute Kalman Gain | $K_{t} = P_{t\\|t-1}Z^{T}(ZP_{t\\|t-1}Z^{T} + H_{t})^{-1} \\quad (5)$ |\n",
-    "|Estimate current state vector| $\\hat{x}_{t\\|t} = \\hat{x}_{t\\|t-1} + K_{t}(y_{t} - Z\\hat{x}_{t\\|t-1}) \\quad (6)$ |\n",
-    "|Estimate current state/process covariance| $P_{t\\|t} = (I - K_{t}Z_{t})P_{t\\|t-1}(I - K_{t}Z_{t})^{T} + K_{t}H_{t}K_{t}^{T} \\quad (7)$ |\n",
+    "|Predict the next state vector| $\\hat{x}_{t+1\\|t} = A_{t}\\hat{x}_{t\\|t}$ |\n",
+    "|Predict the next state/process covariance| $P_{t+1\\|t} = A_{t}P_{t+1\\|t}A_{t}^{T} + Q$ |\n",
+    "|Compute Kalman Gain | $K_{t} = P_{t\\|t-1}Z^{T}(ZP_{t\\|t-1}Z^{T} + H_{t})^{-1}$ |\n",
+    "|Estimate current state vector| $\\hat{x}_{t\\|t} = \\hat{x}_{t\\|t-1} + K_{t}(y_{t} - Z\\hat{x}_{t\\|t-1})$ |\n",
+    "|Estimate current state/process covariance| $P_{t\\|t} = (I - K_{t}Z_{t})P_{t\\|t-1}(I - K_{t}Z_{t})^{T} + K_{t}H_{t}K_{t}^{T}$ |\n",
     "\n",
     ":::{note}\n",
     "We wrote the equation for $P_{t\\|t}$ above using Joseph form, which is more numerically stable but also wordier. In different texts you may encounter this equation written in \"standard\" form.\n",

examples/case_studies/ssm_hurricane_tracking.myst.md

@@ -28,13 +28,14 @@ myst:
 In this case study we are going to forecast the paths of hurricanes by applying several State Space Models (SSM). We will begin with a simple two-dimensional constant acceleration tracking model, where we only have one parameter to estimate. Subsequently, we will progressively add complexity and parameters as we develop up our model. 
 
 As a brief introduction to SSMs, the general idea is that we define our system using two equations.<br> 
-The state equation [1] and the observation equation [2]. 
+The state equation and the observation equation. 
+
 $$
-x_{t+1} = A_{t}x_{t} + c_{t} + R_{t}\epsilon_{t} \quad (1)
+x_{t+1} = A_{t}x_{t} + c_{t} + R_{t}\epsilon_{t}
 $$ 
 
 $$
-y_{t} = Z_{t}x_{t} + d_{t} + \eta_{t} \quad (2)
+y_{t} = Z_{t}x_{t} + d_{t} + \eta_{t}
 $$
 
 The process/state covariance is given by $\epsilon_{t} \sim N(0, Q_{t})$ where $Q_{t}$ is the process/state innovations and the observation/measurement covariance is given by $\eta_{t} \sim N(0, H_{t})$ where $H_{t}$ describes the uncertainty in the measurement device or measurement procedure. 
@@ -69,11 +70,11 @@ The general idea is that we make predictions based on our current state vector a
 The following equations define the process:
 |Description|Equation|
 | --- | --- |
-|Predict the next state vector| $\hat{x}_{t+1\|t} = A_{t}\hat{x}_{t\|t} \quad (3)$ |
-|Predict the next state/process covariance| $P_{t+1\|t} = A_{t}P_{t+1\|t}A_{t}^{T} + Q \quad (4)$ |
-|Compute Kalman Gain | $K_{t} = P_{t\|t-1}Z^{T}(ZP_{t\|t-1}Z^{T} + H_{t})^{-1} \quad (5)$ |
-|Estimate current state vector| $\hat{x}_{t\|t} = \hat{x}_{t\|t-1} + K_{t}(y_{t} - Z\hat{x}_{t\|t-1}) \quad (6)$ |
-|Estimate current state/process covariance| $P_{t\|t} = (I - K_{t}Z_{t})P_{t\|t-1}(I - K_{t}Z_{t})^{T} + K_{t}H_{t}K_{t}^{T} \quad (7)$ |
+|Predict the next state vector| $\hat{x}_{t+1\|t} = A_{t}\hat{x}_{t\|t}$ |
+|Predict the next state/process covariance| $P_{t+1\|t} = A_{t}P_{t+1\|t}A_{t}^{T} + Q$ |
+|Compute Kalman Gain | $K_{t} = P_{t\|t-1}Z^{T}(ZP_{t\|t-1}Z^{T} + H_{t})^{-1}$ |
+|Estimate current state vector| $\hat{x}_{t\|t} = \hat{x}_{t\|t-1} + K_{t}(y_{t} - Z\hat{x}_{t\|t-1})$ |
+|Estimate current state/process covariance| $P_{t\|t} = (I - K_{t}Z_{t})P_{t\|t-1}(I - K_{t}Z_{t})^{T} + K_{t}H_{t}K_{t}^{T}$ |
 
 :::{note}
 We wrote the equation for $P_{t\|t}$ above using Joseph form, which is more numerically stable but also wordier. In different texts you may encounter this equation written in "standard" form.