Added error analysis to mathematical physics.

config/en/mkdocs.yaml

@@ -154,6 +154,10 @@ nav:
         - 'Polarisation': physics/electromagnetism/optics/polarisation.md
 #   - 'Statistical physics':
     - 'Mathematical physics': 
+      - 'Error analysis': 
+        - 'Error analysis formalism': physics/mathematical-physics/error-analysis/formalism.md
+        - 'Maximum error': physics/mathematical-physics/error-analysis/maximum-error.md
+        - 'Standard error': physics/mathematical-physics/error-analysis/standard-error.md
       - 'Signal analysis': 
         - 'Signals': physics/mathematical-physics/signal-analysis/signals.md
         - 'Fourier series': physics/mathematical-physics/signal-analysis/fourier-series.md

docs/en/physics/mathematical-physics/error-analysis/formalism.md

@@ -0,0 +1,32 @@
+# Formalism of error analysis
+
+When measuring a physical quantity with an experiment it is key to know how accurate the physical quantity has been determined, or equivalently, what the uncertainty is in the measured value.
+
+## Measurement errors
+
+Experimental uncertainties that cause a difference between the measured value and the real value of a physical quantity can be grouped into two categories; the **random error** and the **systematic error**. 
+
+Systematic errors always give an error in the same direction when the experiment is repeated. Whereas random errors have no preferential direction when the experiment is repeated. 
+
+## Confidence intervals
+
+The uncertainty in the measured value may be expressed in a **confidence interval**. We will distinguish between two kinds of confidence intervals, the **maximum error** or 100% confidence interval and the **standard error** or 68% confidence interval. The percentage corresponding to this confidence interval is the probability that the real value lies within this interval.
+
+### The maximum error
+
+When a measurement is performed in which all systematic errors have been eliminated and no random errors are observed the maximum error should be used. Additionaly, the maximum error should be used for experiments where only a single measurement has been performed.
+
+When the maximum error is used it is self-evident that multiple measurements of the same quantity are consistent if their confidence intervals overlap. 
+
+### The standard error
+
+The standard error should be used whenever random errors in the measurements are present and when more than one measurement is performed. The standard error may then be determined from the spread in the results.
+
+## Conventions
+
+The following conventions are in use to denote uncertainties.
+
+1. Uncertainties in the measurement results will be denoted with one significant figure, rounding is necessary. For intermediate results, two significant figures can be taken into account.
+2. The least significant figure in a result has to have the same position as that of the uncertainty.
+3. Units have to be mentioned and both the results and the uncertainty should obviously have the same unit.
+4. Uncertainties are always positive.

docs/en/physics/mathematical-physics/error-analysis/maximum-error.md

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+# The maximum error
+
+## Determining the transformed maximum error
+
+In this section a method will be postulated and derived under certain assumptions to determine the maximum error, after a transformation with a map $f$. 
+
+> *Definition 1*: let $f: \mathbb{R}^n \to \mathbb{R} :(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \mapsto f(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \overset{.} = y \pm \Delta_y$ be a function that maps independent measurements with a corresponding maximum error to a new quantity $y$ with maximum error $\Delta_y$ for $n \in \mathbb{N}$. 
+
+In assumption that the maximum errors of the independent measurements are small the following may be posed. 
+
+> *Postulate 1*: let $f:(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \mapsto f(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \overset{.} = y \pm \Delta_y$, the maximum error $\Delta_y$ may be given by
+>
+> $$
+>   \Delta_y = \sum_{i=1}^n | \partial_i f(x_1, \dots, x_n) | \Delta_{x_i}, 
+> $$
+>
+> and $y = f(x_1, \dots, x_n)$ correspondingly for all $(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \in \mathbb{R}^n$. 
+
+??? note "*Derivation*:"
+
+    Will be added later.
+
+With this general expression the following properties may be derived.
+
+### Properties
+
+The sum of the independently measured quantities is posed in the following corollary. 
+
+> *Corollary 1*: let $f:(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \mapsto f(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \overset{.} = y \pm \Delta_y$ with $y$ given by
+>
+> $$
+>   y = f(x_1, \dots, x_n) = x_1 + \dots x_n,
+> $$
+>
+> then the maximum error $\Delta_y$ may be given by
+>
+> $$
+>   \Delta_y = \Delta_{x_1} + \dots + \Delta_{x_n},
+> $$
+>
+> for all $(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \in \mathbb{R}^n$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+The multiplication of a constant with the independently measured quantities is posed in the following corollary. 
+
+> *Corollary 2*: let $f:(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \mapsto f(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \overset{.} = y \pm \Delta_y$ with $y$ given by
+>
+> $$
+>   y = f(x_1, \dots, x_n) = \lambda(x_1 + \dots x_n),
+> $$
+>
+> for $\lambda \in \mathbb{R}$ then the maximum error $\Delta_y$ may be given by
+>
+> $$
+>   \Delta_y = |\lambda| (\Delta_{x_1} + \dots + \Delta_{x_n}),
+> $$
+>
+> for all $(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \in \mathbb{R}^n$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+The product of two independently measured quantities is posed in the following corollary.
+
+> *Corollary 3*: let $f: (x_1 \pm \Delta_{x_1}, x_2 \pm \Delta_{x_2}) \mapsto f(x_1 \pm \Delta_{x_1}, x_2 \pm \Delta_{x_2}) \overset{.} = y \pm \Delta_y$ with $y$ given by
+>
+> $$
+>   y = f(x_1, x_2) = x_1 x_2,
+> $$
+>
+> then the maximum error $\Delta_y$ may be given by
+>
+> $$
+>   \Delta_y = \frac{\Delta_{x_1}}{|x_1|} + \frac{\Delta_{x_2}}{|x_2|}, 
+> $$    
+>
+> for all $(x_1 \pm \Delta_{x_1}, x_2 \pm \Delta_{x_2}) \in \mathbb{R}^2$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Combining measurements
+
+If by a measurement series $m \in \mathbb{N}$ values $\{y_1 \pm \Delta_{y_1}, \dots, y_m \pm \Delta_{y_m}\}$ have been found for the same quantity then 
+
+$$
+    [y \pm \Delta_y] = \bigcap_{i \in \mathbb{N}[i \leq m]} [y_i \pm \Delta_{y_i}],
+$$
+
+the overlap of all the intervals with $[y \pm \Delta_y]$ denoting the interval in which the real value exists.

docs/en/physics/mathematical-physics/error-analysis/standard-error.md

@@ -0,0 +1,149 @@
+# Standard error
+
+## The spread in the mean
+
+> *Definition 1*: for a series of $N \in \mathbb{N}$ independent measurements $\{x_1, \dots, x_N\}$ of the same quantity, the mean $\bar x$ of the measurements is defined as
+>
+> $$
+>   \bar x = \frac{1}{N} \sum_{i=1}^N x_i,
+> $$
+>
+> for all $x_i \in \mathbb{R}$. 
+
+??? note "*Derivation from the expectation value*:"
+
+    Will be added later.
+
+Which is closely related to the expectation value defined in [probability theory](), the difference is the experimental notion of a finite amount of measurements. Similarly, the mean should be an approximation of the true value.
+
+> *Definition 2*: for a series of $N \in \mathbb{N}$ independent measurements $\{x_1, \dots, x_N\}$ of the same quantity, the spread $S$ in the measurements is defined as
+>
+> $$
+>   S = \sqrt{\frac{1}{N - 1} \sum_{i=1}^N (\bar x - x_i)^2},
+> $$
+>
+> for all $x_i \in \mathbb{R}$. 
+
+??? note "*Derivation from the variance*:"
+
+    Will be added later.
+
+Which is closely related to the variance defined in [probability theory](), the difference is once again the experimental notion of a finite amount of measurements.
+
+With the spread $S$ in the measurements the spread in the mean $S_{\bar x}$ may be determined. 
+
+> *Theorem 1*: for a series of $N \in \mathbb{N}$ independent measurements $\{x_1, \dots, x_N\}$ of the same quantity, the spread in the mean $S_{\bar x}$ is given by
+>
+> $$
+>   S_{\bar x} = \sqrt{\frac{1}{N(N-1)} \sum_{i=1}^N (\bar x - x_i)^2}, 
+> $$
+>
+> for all $x_i \in \mathbb{R}$ with $\bar x$ the mean.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Determining the transformed spread
+
+In this section a method will be postulated and derived under certain assumptions to determine the spread in the transformed means with a map $f$. 
+
+> *Definition 3*: let $f: \mathbb{R}^n \to \mathbb{R} :(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \mapsto f(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \overset{.} = \bar y \pm S_{\bar y}$ be a function that maps the mean for each independent measurement series with a corresponding spread to a new mean quantity $\bar y$ with a spread $S_{\bar y}$ for $n \in \mathbb{N}$. 
+
+In assumption that the spread in the mean for each independent measurement series is small, the following may be posed.
+
+> *Postulate 1*: let $f: (\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \mapsto f(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \overset{.} = \bar y \pm S_{\bar y}$, the spread $S_{\bar y}$ may be given by
+>
+> $$
+>   S_{\bar y} = \sqrt{\sum_{i=1}^n \Big(\partial_i f(\bar x_1, \dots, \bar x_n) S_{\bar x_i} \Big)^2},
+> $$
+>
+> and $\bar y = f(\bar x_1, \dots, \bar x_n)$ correspondingly for all $(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \in \mathbb{R}^n$. 
+
+??? note "*Derivation*:"
+
+    Will be added later.
+
+With this general expression the following properties may be derived.
+
+### Properties
+
+The sum of the independently measured quantities is posed in the following corollary. 
+
+> *Corollary 1*: let $f: (\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \mapsto f(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \overset{.} = \bar y \pm S_{\bar y}$ with $\bar y$ given by
+>
+> $$
+>   \bar y = f(\bar x_1, \dots, \bar x_n) = \bar x_1 + \dots \bar x_n,
+> $$
+>
+> then the spread $S_{\bar y}$ may be given by
+>
+> $$
+>   S_{\bar y} = \sqrt{S_{\bar x_1}^2 + \dots + S_{\bar x_n}^2},
+> $$
+>
+> for all $(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \in \mathbb{R}^n$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+The multiplication of a constant with the independently measured quantities is posed in the following corollary. 
+
+> *Corollary 2*: let $f: (\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \mapsto f(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \overset{.} = \bar y \pm S_{\bar y}$ with $\bar y$ given by
+>
+> $$
+>   \bar y = f(\bar x_1, \dots, \bar x_n) = \lambda(\bar x_1 + \dots \bar x_n),
+> $$
+>
+> for $\lambda \in \mathbb{R}$ then the spread $S_{\bar y}$ may be given by
+>
+> $$
+>   S_{\bar y} = |\lambda| (S_{\bar x_1} + \dots + S_{\bar x_n}),
+> $$
+>
+> for all $(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \in \mathbb{R}^n$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+The product of two independently measured quantities is posed in the following corollary.
+
+> *Corollary 3*: let $f: (\bar x_1 \pm S_{\bar x_1}, \bar x_2 \pm S_{\bar x_2}) \mapsto f(\bar x_1 \pm S_{\bar x_1}, \bar x_2 \pm S_{\bar x_2}) \overset{.} = \bar y \pm S_{\bar y}$ with $\bar y$ given by
+>
+> $$
+>   \bar y = f(\bar x_1, \bar x_2) = \bar x_1 \bar x_2,
+> $$
+>
+> then the spread $S_{\bar y}$ may be given by
+>
+> $$
+>   S_{\bar y} = \sqrt{\bigg(\frac{S_{\bar x_1}}{\bar x_1}\bigg)^2 + \bigg(\frac{S_{\bar x_2}}{\bar x_2} \bigg)^2}, 
+> $$    
+>
+> for all $(\bar x_1 \pm S_{\bar x_1}, x_2 \pm S_{\bar x_2}) \in \mathbb{R}^2$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Combining measurements
+
+If by a measurement series $m \in \mathbb{N}$ values $\{\bar y_1 \pm S_{\bar y_1}, \dots, \bar y_m \pm S_{\bar y_m}\}$ have been found for the same quantity then $\bar y$ is given by
+
+$$
+    \bar y = \frac{\sum_{i=1}^m (1 / S_{\bar y_i})^2 \bar y_i}{\sum_{i=1}^m (1 / S_{\bar y_i})^2},
+$$
+
+with its corresponding spread $S_{\bar y}$ given by
+
+$$
+    S_{\bar y} = \frac{1}{\sqrt{\sum_{i=1}^m (1 / S_{\bar y_i})^2}},
+$$
+
+for all $\{\bar y_1 \pm S_{\bar y_1}, \dots, \bar y_m \pm S_{\bar y_m}\} \in \mathbb{R}^m$. 
+
+??? note "*Proof*:"
+
+    Will be added later.