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Question

I would like to have a function: $$ f: \mathbb{R} \rightarrow \left\{x \in \mathbb{R}^n \mid \sum_{i=1}^n x_i = 1\right\}:x \mapsto (f_1(x),\dots,f_n(x)) $$ s.t. $f(0) = (1,0,\dots,0)$ and $f(1) = (0,\dots,0,1)$. This is of course very easy to find but I would like to have these such that the conversion happens in a smooth way. What I mean by that is that I would like to have the following extra properties:

  • $f_1(1/2) = \dots = f_n(1/2) = 1/n$
  • $f_1(x)$ is descending
  • $f_n(x)$ is ascending
  • If we define the function $g_x(k) := f_k(x)$ and we draw the graphs of $g_x(k)$ for $x$ going from $0$ to $1$ this should look like a bit like a wave (idea further explained in next two points)
  • On $]0,1/2[$ we have $f_1(x) > \dots > f_n(x)$ and on $]1/2,1[$ we have $f_1(x) < \dots < f_n(x)$

Example method

For example for $n = 2$ we can define: $$ f(x) := (x, 1-x) $$ now for $n = 3$ we can write: $$ f(x) := (x,f_2(x),1-x) $$ but then we have a problem as we need to have $f_2(x) = 0$, therefore we need to let $f_1(x)$ descend to $1/3$ on $[0,1/2]$ and $f_3(x)$ ascend to $1/3$ on $[0,1/2]$ we define: $$ f_1(x) :=\begin{cases} (1 - \frac{4}{3} \cdot x) & \mbox{ if } x \in [0,1/2]\\ \frac{2}{3}(1 - x) & \mbox{ if } x \in [1/2,1] \end{cases} $$ and $$ f_3(x) :=\begin{cases} \frac{2}{3} \cdot x & \mbox{ if } x \in [0,1/2]\\ \frac{4}{3} x - \frac{1}{3} & \mbox{ if } x \in [1/2,1] \end{cases} $$ then we automatically find for $f_2(x)$: $$ f_2(x) = \begin{cases} \frac{2}{3} x & \mbox{ if } x \in [0,1/2]\\ \frac{2}{3}(1 - x) & \mbox{ if } x \in [1/2,1] \end{cases} $$

Now we can continue for $n = 4$, the logical way to continue would be to define $f_1(x)$ to be $(1 - \frac{3}{2} x)$ on $[0,1/2]$ and $f_4(x) := \frac{1}{2} x$ on $[0,\frac{1}{2}]$. We could then use $f_2 = f_3$ and solve the equation $f_1 + 2 f_2 + f_4 = 1$ on $[0,\frac{1}{2}]$. But this does not satisfy the last property and doesn't look enough like a wave to me.

Reason For My Question

The Reason I ask this is because I would like to have a sequence of Markov Chains which goes from perfect correlation (i.e. transition matrix the unity matrix) to some other Markov Chain which is strongly negatively correlated.

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  • $\begingroup$ Do you need $\sum f_i(x)=1$ always, or just at $x=0,{1\over2},1$? $\endgroup$ Jan 19, 2017 at 19:24
  • $\begingroup$ Always, I'll add the reason for my qustion which might make things more clear. $\endgroup$
    – HolyMonk
    Jan 19, 2017 at 19:25
  • $\begingroup$ Some comments: I think you want $f_1$ descending (decreasing) and $f_n$ ascending (increasing). Also, due to the symmetry of the problem, you can simplify some of the requirements, by looking for a function $f:[0,1/2]\to\mathbb{R}^n$ satisfying the required properties on the half-interval, then for $t\in(1/2,1)$ put $f_k(t)=f_{n+1-k}(1-t)$ for $k=1,\ldots,n$. $\endgroup$
    – Aweygan
    Jan 19, 2017 at 19:30
  • $\begingroup$ I fully agree, made the changes to the question. $\endgroup$
    – HolyMonk
    Jan 19, 2017 at 19:31

1 Answer 1

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$$f_k(x)={q^{k-1}\over\sum\limits_{i=0}^{n-1}q^i},\text{ where }q={x\over1-x}$$ would do the job.

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  • $\begingroup$ Perfect, I plotted it and it's the wave I was looking for, thanks! $\endgroup$
    – HolyMonk
    Jan 19, 2017 at 19:55

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