sections: write a lot

slides/sections/0.md

@@ -1,5 +1,5 @@
 ---
-title: Randomness tests of a non-uniform distribution
+title: Title
 date: \today
 author:
   - Giulia Marcer
@@ -45,9 +45,24 @@ header-includes: |
     \hspace{30pt} \Longrightarrow \hspace{30pt}
   }
 
+  % "and" in formulas
+  \DeclareMathOperator{\et}{%
+    \hspace{30pt} \wedge \hspace{30pt}
+  }
+
   % "with" in formulas
   \DeclareMathOperator{\with}{%
   \hspace{30pt} \text{with} \hspace{30pt}
   }
+
+  % "expected" in formulas
+  \DeclareMathOperator{\ex}{%
+    ^{\text{exp}}
+  }
+
+  % "observed" in formulas
+  \DeclareMathOperator{\ob}{%
+    ^{\text{obs}}
+  }
   ```
 ...

slides/sections/1.md

@@ -3,20 +3,24 @@
 
 ## Goal
 
-What?
+- Generate a sample $L$ of points from a Landau PDF
+- Generate a sample $M$ of points from a Moyal PDF
 
-- Generate a sample of points from a Moyal PDF
+. . .
-- Prove it truly comes from it and not from a Landau PDF
 
-How?
+- Implement a bunch of statistical tests
 
-- Applying some hypothesis testings  
+. . .
+
+- Check if they work:
+  - the sample $L$ truly comes from a Landau PDF
+  - the sample $M$ does not come from a Landau PDF
 
 
 ## Why?
 
-The Landau and Moyal PDFs are really similar. Historically, the latter distribution was utilized in
+The Landau and Moyal PDFs are really similar. Historically, the latter was
-the approximation of the Landau Distribution.
+utilized in the approximation of the former.
 
 :::: {.columns}
 :::  {.column width=33%}
@@ -53,6 +57,8 @@ the approximation of the Landau Distribution.
 :::
 ::::
 
+\vspace{10pt}
+
 :::: {.columns}
 :::  {.column width=50%}
   ![](images/landau-pdf.pdf)
@@ -63,6 +69,25 @@ the approximation of the Landau Distribution.
 :::
 ::::
 
+
 ## Two similar distributions
 
 ![](images/both-pdf.pdf)
+
+
+## Statistical tests
+
+. . .
+
+- Parameters comparison:
+  - compatibility between expected and observed PDF parameters
+
+. . .
+
+- Kolmogorov - Smirnov:
+  - compatibility between expected and observed CDF
+
+. . . 
+
+- Trapani test:
+  - compatibiity between expected and observed mean

slides/sections/2.md

@@ -12,7 +12,7 @@ Because of its fat tail:
 
 . . .
 
-No closed form for parameters.
+No closed form for parameters $\thus$ Numerical estimations
 
 
 ## Landau median
@@ -20,38 +20,48 @@ No closed form for parameters.
 The median of a PDF is defined as:
 
 $$
-  Q_L(m) = \frac{1}{2}
+  m = Q \left( \frac{1}{2} \right)
 $$
 
 . . .
 
-- CDF computed by numerical integration,
+- CDF computed by numerical integration
 - QDF computed by numerical root-finding (Brent)
 
 $$
-  m_L = 1.3557804...
+  m_L\ex = 1.3557804...
 $$
 
 
 ## Landau mode
 
-- Maxmimum $\quad \Longrightarrow \quad \partial_x M(\mu) = 0$,
+- Maxmimum $\quad \Longrightarrow \quad \partial_x L(\mu) = 0$
+
+. . .
+
 - Computed by numerical minimization (Brent)
 
 $$
-  \mu_L = − 0.22278...
+  \mu_L\ex = − 0.22278...
 $$
 
 
 ## Landau FWHM
 
-$$
+We need to compute the maximum:
-  \text{FWHM} = x_+ - x_- \with L(x_{\pm})
-              = \frac{L_{\text{max}}}{2} = \frac{L(\mu_L)}{2}
-$$
-
-- Computed numerically (Brent)
 
 $$
-  \text{FWHM}_L = 4.018645...
+  L_{\text{max}} = L(\mu_L)
+$$
+
+$$
+  \text{FWHM} = w = x_+ - x_- \with L(x_{\pm}) = \frac{L_{\text{max}}}{2}
+$$
+
+. . .
+
+- Computed by numerical root finding (Brent)
+
+$$
+  w_L\ex = 4.018645...
 $$

slides/sections/3.md

@@ -50,7 +50,7 @@ $$
 
 Remembering the error function
 $$
-  \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x dy \, e^{-y^2},
+  \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x dy \, e^{-y^2}
 $$
 one finally gets:
 $$
@@ -72,14 +72,14 @@ $$
 
 ## Moyal median
 
-Defined by $\text{CDF}(m) = 1/2$, or $m=\text{QDF}(1/2)$.
+Defined by $F(m) = \frac{1}{2}$ or $m = Q \left( \frac{1}{2} \right)$:
 
 \begin{align*}
   M(z)
-    &\thus m_M = -2 \ln \left[ \sqrt{2} \,
+    &\thus m_M\ex = -2 \ln \left[ \sqrt{2} \,
     \text{erf}^{-1} \left( \frac{1}{2} \right) \right] \\
   M_{\mu \sigma}(x)
-    &\thus m_M = \mu -2 \sigma \ln \left[ \sqrt{2} \,
+    &\thus m_M\ex = \mu -2 \sigma \ln \left[ \sqrt{2} \,
     \text{erf}^{-1} \left( \frac{1}{2} \right) \right]
 \end{align*}
 
@@ -95,8 +95,8 @@ $$
 $$
 
 \begin{align*}
-  \partial_x M(z) = 0 &\thus \mu_M = 0                \\
+  \partial_x M(z) = 0 &\thus \mu_M\ex = 0                \\
-  \partial_x M_{\mu \sigma}(x) = 0 &\thus \mu_M = \mu \\
+  \partial_x M_{\mu \sigma}(x) = 0 &\thus \mu_M\ex = \mu \\
 \end{align*}
 
 
@@ -129,7 +129,7 @@ $$
 
 \begin{align*}
   M(z)
-    &\thus \text{FWHM}_M = a              \\
+    &\thus w_M^{\text{exp}} = a              \\
   M_{\mu \sigma}(x)
-    &\thus \text{FWHM}_M = \sigma \cdot a \\
+    &\thus w_M^{\text{exp}} = \sigma \cdot a \\
 \end{align*}

slides/sections/4.md

@@ -1,35 +1,86 @@
-# Data sample
+# Sample parameters estimation
 
 
-## PDF parameters
+## Sample parameters estimation
 
-A $M(x)$ similar to $L(x)$ can be found by imposing:
+Once the points are sampled, how to estimate their median, mode and FWHM?
+
+. . .
+
+- Binning data $\quad \longrightarrow \quad$ result depending on bin-width
+
+. . .
+
+- Alternative solutions
+
+
+## Sample median
 
-- equal mode
 $$
-  \mu_M = \mu_L \approx −0.22278298...
+  m = Q \left( \frac{1}{2} \right)
-$$
-
-- equal width
-$$
-  \text{FWHM}_M = \text{FWHM}_L = \sigma \cdot a
 $$
 
 . . .
 
+- Sort points in ascending order
+
+. . .
+
+- Middle element if odd
+- Average of the two central elements if even
+
+
+## Sample mode
+
+Most probable value
+
+. . .
+
+HSM
+
+- Iteratively identify the smallest interval containing half points
+- once the sample is reduced to less than three points, take average
+
+
+## Sample FWHM
+
 $$
-  \implies \sigma_M \approx 1.1191486
+  \text{FWHM} = x_+ - x_- \with L(x_{\pm}) = \frac{L_{\text{max}}}{2}
 $$
 
+. . .
 
-## PDF parameters
+KDE
 
-:::: {.columns}
+- empirical PDF construction:
-:::  {.column width=50%}
-  ![](images/both-pdf-bef.pdf)
-:::
 
-:::  {.column width=50%}
+$$
-  ![](images/both-pdf-aft.pdf)
+  f_\varepsilon(x) = \frac{1}{N\varepsilon} \sum_{i = 1}^N
-:::
+  G \left( \frac{x-x_i}{\varepsilon} \right)
-::::
+$$
+
+The parameter $\varepsilon$ controls the strenght of the smoothing
+
+
+## Sample FWHM
+
+Silverman's rule of thumb:
+
+$$
+  f_\varepsilon(x) = \frac{1}{N\varepsilon} \sum_{i = 1}^N
+  G \left( \frac{x-x_i}{\varepsilon} \right)
+  \with
+  \varepsilon = 0.63 \, S_N
+  \left( \frac{d + 2}{4}N \right)^{-1/(d + 4)}
+$$
+
+with:
+
+- $S_N$ is the sample stdev
+- $d$ number of dimensions ($d = 1$)
+
+. . .
+
+\vspace{10pt}
+
+Numerical root finding (Brent)

slides/sections/5.md

@@ -0,0 +1,66 @@
+# MC simulations
+
+
+## In summary
+
+-----------------------------------------------------
+                  Landau            Moyal
+----------------- ----------------- -----------------
+median             $m_L\ex$          $m_M\ex (μ, σ)$
+
+mode               $\mu_L\ex$        $\mu_M\ex (μ)$
+
+FWHM               $w_L\ex$          $w_M\ex (σ)$
+-----------------------------------------------------
+
+
+## PDF parameters
+
+A $M(x)$ similar to $L(x)$ can be found by imposing:
+
+\vspace{15pt}
+
+- equal mode
+$$
+  \mu_M\ex = \mu_L\ex \approx −0.22278298...
+$$
+
+. . .
+
+- equal width
+$$
+  w_M\ex = w_L\ex = \sigma \cdot a
+$$
+
+$$
+  \implies \sigma_M \approx 1.1191486
+$$
+
+
+## PDF parameters
+
+:::: {.columns}
+:::  {.column width=50%}
+  ![](images/both-pdf-bef.pdf)
+:::
+
+:::  {.column width=50%}
+  ![](images/both-pdf-aft.pdf)
+:::
+::::
+
+
+## Different medians
+
+This leads to more different medians:
+
+\begin{align*}
+  m_M = 0.787... \thus &m_M = 0.658... \\
+                       &m_L = 1.355...
+\end{align*}
+
+
+## Samples
+
+- Sample $L$: N = 50'000 points following $L_(x)$
+- Sample $M$: N = 50'000 points following $M_{\mu \sigma}(x)$