For a problem I am working on, I came across the data shown in the table below, that is also displayed in the accompanying graph.
For both the Y1 and Y2 data plotted vs. the X value it looks like some type of hyperbola with a horizontal asymptote at 1 and a vertical asymptote at around 898, but because I am not that mathematically skilled, I can't seem to figure out what kind(s) of function(s) f(x) could be used to describe this data.
X Y1 Y2
253.6407994 1.008716206 1.034755788
373.6998994 1.029072813 1.115300361
504.5643184 1.062562202 1.246710815
627.2366248 1.110882816 1.436932395
725.929416 1.171186528 1.67693405
795.920823 1.242664652 1.96650879
840.5698058 1.323924639 2.302179005
866.7363934 1.411516038 2.670089968
881.2738761 1.502484536 3.057072276
889.1273653 1.596120247 3.459065105
893.2850369 1.691845661 3.872578159
895.4534286 1.789046088 4.294130583
896.5728192 1.888432275 4.726234972
897.1421632 1.989661187 5.167006184
897.4281541 2.092268314 5.614163642
897.5704807 2.197216786 6.071750713
897.6401956 2.302139486 6.5293478
897.6743507 2.40705667 6.986987281
897.6910851 2.511965162 7.444624582
897.6992847 2.616876345 7.902292702
897.7033024 2.721783957 8.359955353
What would be good matches for this type of data?
EDIT
Some additional info after I tried a number of transformations. It appears that for the initial data points (excluding the last 4), a transformation of the X value leads to a pretty decent approximation.
The transformation is: t(x) = ln(1 / (x-limit - x))
When plotted vs. Y data (excluding the last 4), a polynomial f(x) = a * t(x)^2 + b * t(x) + c can be fitted:
Which then describes the initial part of the data rather accurately.
This looks very much like the sort of curves you get when measuring strain up to a failure time $t_f$. A possible motivation for curves of this type is the following second order differential equation $$ \frac{dy^2}{dt^2} - a(\frac{dy}{dt})^\alpha = 0, $$ where $\alpha \neq 1$ creates nonlinear and possibly powerlaw behaviour, this equation has general solution of the form $$ y = y_0\left[1 + \frac{t}{m\tau} \right]^m, \; \; m \neq 0, $$ and in particular in the situation where $m < 0$ we have $$ y = y_0 \left[ 1 - \frac{t}{t_f} \right]^m. $$ Suggesting after a bit of rearranging that a good candidate function for your data would be $$ f(x) = \beta_0 \left[x_f - x \right]^{\beta_1},\; \; \beta_1 < 0. $$ Taking $x_f = 898.0$ and then fitting the remaining parameters to your dataset $\{ X,Y_1 \} $ by least squares I get
Obviously there are improvements to be made but I think that is a reasonable starting off point. For more on the strain rate differential equations have a look at A damage mechanics model for power-law creep and earthquake aftershock and foreshock sequences.