Update documentation

_sources/notebooks/00/resources.md

@@ -1,4 +1,4 @@
-# Resources
+# Resources and acknowledgments
 
 The following resources served as inspiration and provided very useful content to write some of the sections of the book. 
 
@@ -18,4 +18,11 @@ The following resources served as inspiration and provided very useful content t
 - IPython Cookbook, Second Edition (2018) [[Link](https://ipython-books.github.io/121-plotting-the-bifurcation-diagram-of-a-chaotic-dynamical-system/)].
 - Introduction to Taken's Embedding [[link](https://www.kaggle.com/code/tigurius/introduction-to-taken-s-embedding/notebook)].
 - An introduction to Dynamic Time Warping [[link](https://rtavenar.github.io/blog/dtw.html)].
-- An intuitive approach to DTW — Dynamic Time Warping [[link](https://towardsdatascience.com/an-intuitive-approach-to-dtw-dynamic-time-warping-f660ccb77ff4)].
+- An intuitive approach to DTW — Dynamic Time Warping [[link](https://towardsdatascience.com/an-intuitive-approach-to-dtw-dynamic-time-warping-f660ccb77ff4)].
+
+## Acknowledgments
+
+I whish to thank:
+
+- [Simone Scardapane](https://www.sscardapane.it/) for spotting many typos, giving feedback, and suggestions.
+- Jonas Berg Hansen for helping preparing the exercises.

_sources/notebooks/03/smoothing.ipynb

@@ -40,7 +40,7 @@
     "tags": []
    },
    "source": [
-    "- In this lecture, we will build that knowledge and explore another important concept called **smoothing**.\n",
+    "- In this lecture, we will build upon that knowledge and explore another important concept called **smoothing**.\n",
     "- In particular, we will cover:\n",
     "    1. An introduction to smoothing and why it is necessary.\n",
     "    2. Common smoothing techniques.\n",
@@ -299,7 +299,7 @@
     "\n",
     "- The approximation with the mean seems reasonable in this case.\n",
     "- In general we want to measure how far off our estimate is from reality. \n",
-    "- A common way of doing it is by calculating *Mean Squared Error* (MSE) \n",
+    "- A common way of doing it is by calculating the *Mean Squared Error* (MSE) \n",
     "\n",
     "$$MSE = \\frac{1}{T}\\sum_{t=1}^{T} (X(t) - \\hat{X}(t))^2$$\n",
     "\n",
@@ -319,7 +319,7 @@
     "- Let's consider the following example.\n",
     "- Say we have a time series of observed values $X = [0, 1, 3, 2]$.\n",
     "- The predictions given by our model are $\\hat{X} =  [1, 1, 2, 4]$. \n",
-    "- We calculate MSE as: \n",
+    "- We calculate the MSE as: \n",
     "\n",
     "$$(0-1)^{2} + (1-1)^{2} + (3-2)^{2} + (2-4)^{2} = 6$$"
    ]
@@ -351,7 +351,7 @@
     "- The MSE allows to compare different estimates to see which is best. \n",
     "- In this case, the first model gives us a better estimate than the second one. \n",
     "- This idea of measuring how a model performs is important in machine learning and we will use it often in this course.\n",
-    "- Let's create a function to calculate MSE that we will use in as we go forward."
+    "- Let's create a function to calculate MSE that we will use as we go forward."
    ]
   },
   {
@@ -684,7 +684,7 @@
    },
    "source": [
     "- Note that the output of the MA is shorter than the original data.\n",
-    "- The reason is because the moving window is not centered on the first and last elements of the time series.\n",
+    "- The reason is that the moving window is not centered on the first and last elements of the time series.\n",
     "- Given a window of size $P$, the MA will be $P-1$ steps shorter than the original time series."
    ]
   },
@@ -719,7 +719,7 @@
     "tags": []
    },
    "source": [
-    "- Now, we try the MA on a period time series, which could represent a seasonality."
+    "- Now, we try the MA on a periodic time series, which could represent a seasonality."
    ]
   },
   {
@@ -777,7 +777,7 @@
    },
    "source": [
     "- It's not perfect but clearly picks up the periodic pattern.\n",
-    "- Lastly, let's see how moving average handles trend, seasonality, and a bit of noise."
+    "- Lastly, let's see how MA handles trend, seasonality, and a bit of noise."
    ]
   },
   {
@@ -898,15 +898,29 @@
     "tags": []
    },
    "source": [
-    "- There are many ways to create the weights.\n",
-    "- A simple way is to define them as follows:\n",
+    "- There are many ways to set the weights.\n",
+    "- For example, we could define them with the following system of equations: \n",
     "\n",
     "$$\n",
+    "\\begin{cases}\n",
+    "& w_1 + w_2 + w_3 = 1\\\\ \n",
+    "& w_2 = (w_1)^2 \\\\ \n",
+    "& w_3 = (w_1)^3\n",
+    "\\end{cases}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "- This gives the following weights:\n",
+    "  \n",
+    "$$\n",
     "\\begin{aligned}\n",
-    "& w + w^2 + w^3 = 1\\\\ \n",
-    "& w_1 = w \\approx 0.543 \\\\ \n",
-    "& w_2 = w^2 \\approx 0.294 \\\\ \n",
-    "& w_3 = w^3 \\approx 0.16 \n",
+    "& w_1 \\approx 0.543 & \\text{weights associated to $t-1$}\\\\ \n",
+    "& w_2 \\approx 0.294 & \\text{weights associated to $t-2$} \\\\ \n",
+    "& w_3 \\approx 0.16 & \\text{weights associated to $t-3$}\n",
     "\\end{aligned}\n",
     "$$\n",
     "\n",
@@ -939,7 +953,7 @@
     "### Forecasting with MA\n",
     "\n",
     "- Instead of pulling out the inherent pattern within a series, the smoothing functions can be used to create *forecasts*. \n",
-    "- The forecast for the next time step is compute as follow \n",
+    "- The forecast for the next time step is computed as follows:\n",
     "\n",
     "$$\\hat{X}(t+1) = \\frac{X(t) + X(t-1) + \\dots + X(t-P+1)}{P}$$ \n",
     "\n",
@@ -1056,7 +1070,7 @@
     "- One might reduce the window size to obtain a more responsive model. \n",
     "- However, a window size that's too small will chase noise in the data as opposed to extracting the pattern. \n",
     "- There is a tradeoff between *responsiveness* and *robustness to noise*. \n",
-    "- The best answer lies somewhere between and requires careful tuning to determine which setup is best for a given dataset and problem at hand. "
+    "- The best answer lies somewhere in between and requires careful tuning to determine which setup is best for a given dataset and problem at hand. "
    ]
   },
   {
@@ -1363,8 +1377,8 @@
     "- There are many initialization strategies. \n",
     "- A simple one is to set $S(0) = X(0)$. \n",
     "- Another strategy is to find the mean of the first couple of observations, e.g., $S(0) = \\frac{X(0) + X(1) + X(2)}{3}$. \n",
-    "- Once you've initialized, you simply use the update rule above to calculate all values. \n",
-    "- In practice, we don't this process by hand, but we use the functions in ``statsmodels``. "
+    "- Once you've initialized it, you can use the update rule above to calculate all values. \n",
+    "- In practice, we don't perform this process by hand, but we use the functions in ``statsmodels``. "
    ]
   },
   {
@@ -1376,10 +1390,10 @@
     "tags": []
    },
    "source": [
-    "#### Setting Alpha\n",
+    "#### Setting $\\alpha$\n",
     "- Choosing the optimal value for $\\alpha$ is also done by ``statsmodels``. \n",
     "- A solver uses a metric like MSE to find the optimal $\\alpha$. \n",
-    "- Even if we do not choose the value manually, is goot to have a basic idea of what's happening under the hood."
+    "- Even if we do not choose the value manually, is good to have a basic idea of what's happening under the hood."
    ]
   },
   {
@@ -1436,7 +1450,7 @@
     "### Triple exponential smoothing\n",
     "\n",
     "- 💡 It has all the benefits of Double Exponential with the ability to also model the seasonality.\n",
-    "- It does by adding a third component that smooths out seasonality of length $L$.\n",
+    "- It does this by adding a third component that smooths out seasonality of length $L$.\n",
     "- There are two variants in the model:\n",
     "    - additive seasonality,\n",
     "    - multiplicative seasonality."
@@ -1452,6 +1466,8 @@
    "source": [
     "**Additive seasonality model**\n",
     "\n",
+    "- This method is preferred when the seasonal variations are roughly constant through the series.\n",
+    "\n",
     "$$\n",
     "\\begin{aligned} \n",
     "S(t) = \\alpha \\big(X(t) + c(t-L)\\big) + (1-\\alpha) \\big(S(t-1) + b(t-1)\\big) & \\hspace{2cm} \\text{Smoothed values} \\\\\n",
@@ -1472,6 +1488,8 @@
    "source": [
     "**Multiplicative seasonality model**\n",
     "\n",
+    "- This method is preferred when the seasonal variations are changing proportional to the level of the series.\n",
+    "\n",
     "$$\n",
     "\\begin{aligned} \n",
     "S(t) = \\alpha \\frac{X(t)}{c(t-L)} + (1-\\alpha) \\big(S(t-1) + b(t-1)\\big) & \\hspace{2cm} \\text{Smoothed values} \\\\\n",
@@ -1493,7 +1511,7 @@
     "**Initialization**\n",
     "\n",
     "- $b(0) = \\frac{1}{L} \\left( \\frac{X(L+1) - X(1)}{L} + \\frac{X(L+2) - X(2)}{L} + \\dots + \\frac{X(L+L) - X(L)}{L} \\right)$\n",
-    "- $c(t) = \\frac{1}{N} \\sum_{j=1}^N \\frac{X(L(j-1)+t)}{A_j}$ for $t=1,2,\\dots,L$, where $N$ is the number of complete cycles in the data and $A_j = \\frac{\\sum_{t=1}^L  X(2(j-1)+t)}{L}$\n",
+    "- $c(t) = \\frac{1}{N} \\sum_{j=1}^N \\frac{X(L(j-1)+t)}{A_j}$ for $t=1,2,\\dots,L$, where $N$ is the total number of seasonal cycles in the data and $A_j = \\frac{\\sum_{t=1}^L  X(2(j-1)+t)}{L}$\n",
     "\n",
     "---"
    ]
@@ -1872,7 +1890,7 @@
    "source": [
     "- Triple Exponential Smoothing pickups trend and seasonality. \n",
     "- Clearly, this is the most suitable approach for this data.\n",
-    "- We can summarize the results in the following"
+    "- We can summarize the results in the following table:"
    ]
   },
   {
@@ -1929,7 +1947,7 @@
     "In this lecture we learned\n",
     "\n",
     "1. What is smoothing and why it is necessary.\n",
-    "2. The common smoothing techniques.\n",
+    "2. Some common smoothing techniques.\n",
     "3. A basic understanding of how to smooth time series data with Python and generate forecasts.\n",
     "---"
    ]