<p>C is purely empirical for any given situation. BEWARE of such equations! Using coefficients to six significant figures gives the <em>illusion</em> of high precision when, in fact, the whole approach is extremely 'rubbery', and highly dependent upon the local geology / soil type / local hydrology. The expression is only valid for the location in which it was calibrated.
A small change in drainage can make a huge change in the equation so, personally, I think such stochastic simplifications of complex processes are not worth the paper they are written on! Intelligent eye-balling the potential landslide conditions, with a view to geology and drainage, may be just as good a guide as any (generally poorly calibrated) equation.</p>

CC BY-SA 3.0
2016-04-17T00:50:51.023Z
I think that there is no issue with dependency of $C$ on location.
sequence
CC BY-SA 3.0
2016-04-16T06:07:13.787Z
2016-04-16T06:07:13.787Z
Gordon Stanger
<p>There are two solutions using the least square method for calculating $C$ and $D$. Both methods yield different results for your constants. There is no <strong>correct</strong> method.</p>

<h2>Method of least Squares</h2>

<p>We define the least square error as follows:
$$\text{lse} = \sum_{i}{\left(y_i - f(x_i)\right)^2}$$
The $y_i$ and $x_i$ are our data through that we want to fit a function $f(x)$. The aim is to reduce our error $\text{lse}$.</p>

<p>The solution (minimal $\text{lse}$) for a linear function $f(x) = a \cdot x + b$ is described <a href="https://en.wikipedia.org/wiki/Linear_least_squares_%28mathematics%29" rel="nofollow noreferrer">here</a>. The basic idea for calculating the minimum of $\text{lse}$ is to set $\partial \text{lse}/\partial a$ and $\partial \text{lse}/\partial b$ equal to $0$ and solve the resulting equation system to $a$ and $b$.</p>

<p>Having said that we come to two solutions for calculating $C$ and $D$ in $f(x) = C\cdot x^{-D}$</p>

<h2>Solution 1: Logarithmize the Function</h2>

<p>We rewrite
$$y_i = C \cdot x_i^{-D}$$
to
$$\ln{y_i} = \ln{\left(C \cdot x_i^{-D}\right)} = \ln{C} - D\cdot \ln{x_i} $$</p>

<p>Now we have a function of the form $\tilde{y_i} = a\cdot \tilde{x_i} + b$ with $\tilde{y_i} = \ln{y_i}$, $\tilde{x_i} = \ln{x_i}$, $a=-D$ and $b=\ln{C}$. Thus, we logarithmize our measured $x_i$ and $y_i$ values and put them into the formulas of the linear least square error method. From the resulting $a$ and $b$ we than calculate $D$ and $C$.</p>

<h2>Solution 2: Insert the Function as it is</h2>

<p>We set $f(x) = C\cdot x^{-D}$, insert it into our $\text{lse}$ formula:
$$ \text{lse} = \sum_i{\left(y_i - C\cdot x^{-D}\right)^2}$$
and minimize the resulting formula with respect to $C$ and $D$ for the given set of $x_i$ and $y_i$. This is a tricky and not as straight-forward as it is in the linear case. You could make $\partial \text{lse}/\partial C$ and $\partial \text{lse}/\partial D$ and look how far you come.</p>

<hr>

<p>I personally, prefer the answer of <a href="https://earthscience.stackexchange.com/users/4507/gordon-stanger">Gordon Stanger</a> :-) .</p>

CC BY-SA 3.0
2016-04-16T19:34:48.440Z
2016-04-16T19:34:48.440Z
2017-04-13T12:50:30.540Z
daniel.heydebreck
<p>Given a distribution obeying the power-law (fractal) relation, such as the cumulative distribution function $L_{cf}(&gt; X) = CR^{-D}$, if $X$ is given, how does one find the constant $C$ from a given data set?</p>

<p>For example, given this table:</p>

<p><a href="https://i.sstatic.net/B5zDG.png"><img src="https://i.sstatic.net/B5zDG.png" alt="Source: C. Li et al. / Geomorphology 130 (2011)"></a></p>

<p>[Source: C. Li et al. / Geomorphology 130 (2011)]</p>

<p>how does one determine how the values for $L_{cf}$ were obtained? It appears that they used the method of least squares fitting to obtain the values for $D$, but I don't seem to understand where the values for $C$ are coming from. Any help would be appreciated.</p>

CC BY-SA 3.0
2016-04-16T11:27:34.960Z
Are you looking for a mathematical method (and you assume that the necessary data is already available)?
daniel.heydebreck
CC BY-SA 3.0
2016-04-16T17:04:11.293Z
That's exactly right.
sequence
<earthquakes><rainfall><statistics><natural-disasters>
How to determine constant $C$ in $p(x) = Cx^{-D}$?
sequence
# How to determine constant $C$ in $p(x) = Cx^{-D}$?

(Asked by: username_0 on 2016-04-16)

Given a distribution obeying the power-law (fractal) relation, such as the cumulative distribution function $L\_{cf}(> X) = CR^{-D}$, if $X$ is given, how does one find the constant $C$ from a given data set?

For example, given this table:

[![Source: C. Li et al. / Geomorphology 130 (2011)](https://i.sstatic.net/B5zDG.png)](https://i.sstatic.net/B5zDG.png)

[Source: C. Li et al. / Geomorphology 130 (2011)]

how does one determine how the values for $L\_{cf}$ were obtained? It appears that they used the method of least squares fitting to obtain the values for $D$, but I don't seem to understand where the values for $C$ are coming from. Any help would be appreciated.

**Tags:** earthquakes, rainfall, statistics, natural-disasters

> **Comments:**
> **username_1**: Are you looking for a mathematical method (and you assume that the necessary data is already available)?  
> **username_0**: That's exactly right.  

---

## 2 answers

### Answer by username_1 on 2016-04-16. Score: 5 | Accepted answer

There are two solutions using the least square method for calculating $C$ and $D$. Both methods yield different results for your constants. There is no **correct** method.

## Method of least Squares

We define the least square error as follows:
$$\text{lse} = \sum\_{i}{\left(y\_i - f(x\_i)\right)^2}$$
The $y\_i$ and $x\_i$ are our data through that we want to fit a function $f(x)$. The aim is to reduce our error $\text{lse}$.

The solution (minimal $\text{lse}$) for a linear function $f(x) = a \cdot x + b$ is described [here](https://en.wikipedia.org/wiki/Linear_least_squares_%28mathematics%29). The basic idea for calculating the minimum of $\text{lse}$ is to set $\partial \text{lse}/\partial a$ and $\partial \text{lse}/\partial b$ equal to $0$ and solve the resulting equation system to $a$ and $b$.

Having said that we come to two solutions for calculating $C$ and $D$ in $f(x) = C\cdot x^{-D}$

## Solution 1: Logarithmize the Function

We rewrite
$$y\_i = C \cdot x\_i^{-D}$$
to
$$\ln{y\_i} = \ln{\left(C \cdot x\_i^{-D}\right)} = \ln{C} - D\cdot \ln{x\_i} $$

Now we have a function of the form $\tilde{y\_i} = a\cdot \tilde{x\_i} + b$ with $\tilde{y\_i} = \ln{y\_i}$, $\tilde{x\_i} = \ln{x\_i}$, $a=-D$ and $b=\ln{C}$. Thus, we logarithmize our measured $x\_i$ and $y\_i$ values and put them into the formulas of the linear least square error method. From the resulting $a$ and $b$ we than calculate $D$ and $C$.

## Solution 2: Insert the Function as it is

We set $f(x) = C\cdot x^{-D}$, insert it into our $\text{lse}$ formula:
$$ \text{lse} = \sum\_i{\left(y\_i - C\cdot x^{-D}\right)^2}$$
and minimize the resulting formula with respect to $C$ and $D$ for the given set of $x\_i$ and $y\_i$. This is a tricky and not as straight-forward as it is in the linear case. You could make $\partial \text{lse}/\partial C$ and $\partial \text{lse}/\partial D$ and look how far you come.

---

I personally, prefer the answer of [username_2 username_2](https://earthscience.stackexchange.com/users/4507/username_2-username_2) :-) .


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### Answer by username_2 on 2016-04-16. Score: 3

C is purely empirical for any given situation. BEWARE of such equations! Using coefficients to six significant figures gives the *illusion* of high precision when, in fact, the whole approach is extremely 'rubbery', and highly dependent upon the local geology / soil type / local hydrology. The expression is only valid for the location in which it was calibrated.
A small change in drainage can make a huge change in the equation so, personally, I think such stochastic simplifications of complex processes are not worth the paper they are written on! Intelligent eye-balling the potential landslide conditions, with a view to geology and drainage, may be just as good a guide as any (generally poorly calibrated) equation.

> **Comments:**
> **username_0**: I think that there is no issue with dependency of $C$ on location.  

---

