Re: SciPy-User Digest, Vol 231, Issue 1

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On Mon, Jan 23, 2023 at 1:29 PM Samuel Dupree <[email protected]> wrote:

> The problem I have is if the interpolator only sees numbers. Unless
> there is some undercover means by which the interpolator sees the
> function used to generate the numbers passed to it, how does it know the
> conversion factor is used argument to the sine function? Furthermore, If
> I were to rewrite the program as follows, I still get the same results.
>
>
>
> import numpy as np
> import matplotlib.pyplot as plt
> from scipy.interpolate import pchip_interpolate
>
> x_observed    = np.linspace(0.0, 360.0, 100)
> xx_observed   = np.pi*x_observed/180
>
> y_observed    = np.sin(xx_observed)
> dydx_observed = np.cos(xx_observed)
>

Note that this is *not* a derivative of `y_observed` with respect to
`x_observed`. It is the derivative with respect to the *transformed
variable* `xx_observed`. If you want the derivative against `x_observed`,
you will have to apply the chain rule and scale the value from `np.cos()`
appropriately.

x    = np.linspace(min(x_observed), max(x_observed), num=100)
> y    = pchip_interpolate(x_observed, y_observed, x, der=0, axis=0)
> dydx = pchip_interpolate(x_observed, y_observed, x, der=1, axis=0)
>

This, on the other hand, is the derivative of `y` with respect to
`x_observed`. Thus, you cannot compare `dydx` directly to `dydx_observed`.
They will differ by the scale factor that you would get from applying the
chain rule correctly.

-- 
Robert Kern

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<div dir=3D"ltr"><div dir=3D"ltr">On Mon, Jan 23, 2023 at 1:29 PM Samuel Du=
pree &lt;<a href=3D"mailto:[email protected]">[email protected]</a>=
&gt; wrote:<br></div><div class=3D"gmail_quote"><blockquote class=3D"gmail_=
quote" style=3D"margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,=
204);padding-left:1ex">The problem I have is if the interpolator only sees =
numbers. Unless <br>
there is some undercover means by which the interpolator sees the <br>
function used to generate the numbers passed to it, how does it know the <b=
r>
conversion factor is used argument to the sine function? Furthermore, If <b=
r>
I were to rewrite the program as follows, I still get the same results.<br>
<br>
<br>
<br>
import numpy as np<br>
import matplotlib.pyplot as plt<br>
from scipy.interpolate import pchip_interpolate<br>
<br>
x_observed=C2=A0=C2=A0=C2=A0 =3D np.linspace(0.0, 360.0, 100)<br>
xx_observed=C2=A0=C2=A0 =3D np.pi*x_observed/180<br>
<br>
y_observed=C2=A0=C2=A0=C2=A0 =3D np.sin(xx_observed)<br>
dydx_observed =3D np.cos(xx_observed)<br></blockquote><div><br></div><div>N=
ote that this is <i>not</i> a derivative of `y_observed` with respect to `x=
_observed`. It is the derivative with respect to the <i>transformed variabl=
e</i> `xx_observed`. If you want the derivative against `x_observed`, you w=
ill have to apply the chain rule and scale the value from `np.cos()` approp=
riately.</div><div><br></div><blockquote class=3D"gmail_quote" style=3D"mar=
gin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1=
ex">
x=C2=A0=C2=A0=C2=A0 =3D np.linspace(min(x_observed), max(x_observed), num=
=3D100)<br>
y=C2=A0=C2=A0=C2=A0 =3D pchip_interpolate(x_observed, y_observed, x, der=3D=
0, axis=3D0)<br>
dydx =3D pchip_interpolate(x_observed, y_observed, x, der=3D1, axis=3D0)<br=
></blockquote><div><br></div><div>This, on the other hand, is the derivativ=
e of `y` with respect to `x_observed`. Thus, you cannot compare `dydx` dire=
ctly to `dydx_observed`. They will differ by the scale factor that you woul=
d get from applying the chain rule correctly.</div><div><br></div></div>-- =
<br><div dir=3D"ltr" class=3D"gmail_signature">Robert Kern</div></div>

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