Download or view curveFit.frink in plain text format
/** This library contains routines to perform a best linear or other type of
fit to a set of data points. This is often referred to as "regression"
because of some historical accident.
These functions are designed to preserve units of measure, to provide
coefficients with correct units of measure, and to work for purely
symbolic values!
To use this file, see the examples in curveFitTest.frink
Those examples are powerful and let you, say, derive the gravitic equation.
Also see LeastSquares.frink and the test program LeastSquaresTest.frink
which allow you to fit arbitrary linear systems to arbitrary basis
functions.
Also see the leastSquares function in Matrix.frink and the test program
MatrixQRTest.frink which demonstrate linear least-squares fitting using QR
decomposition which is general for a wide variety of linear systems with
non-exact measurements.
See:
https://mathworld.wolfram.com/LeastSquaresFitting.html
See also for special curve fits:
TODO: Implement these.
https://mathworld.wolfram.com/LeastSquaresFittingExponential.html
https://mathworld.wolfram.com/LeastSquaresFittingLogarithmic.html
https://mathworld.wolfram.com/LeastSquaresFittingPowerLaw.html
*/
/** Performs a best linear fit of the specified data points. In other words,
this finds the coefficients a and b of a line (in slope-intercept form)
with the equation:
y = a x + b
params:
data: an array or set of [x,y] pairs.
returns:
[a, b, r]
where
r is the correlation coefficient
*/
linearFit[data] :=
{
array = toArray[data]
N = length[array]
if N < 2
return [undef, undef, undef]
// We do it this way to preserve units of measure by initializing the sum
// with the first element.
[x,y] = array@0
sxy = x y // Sum of x*y
sx = x // Sum of x
sy = y // Sum of y
sx2 = x^2 // Sum of x^2
sy2 = y^2 // Sum of y^2
for i = 1 to N-1
{
[x,y] = array@i
sxy = sxy + x y
sx = sx + x
sy = sy + y
sx2 = sx2 + x^2
sy2 = sy2 + y^2
}
denom = N sx2 - sx^2
a = (N sxy - sx sy) / denom
b = (sy sx2 - sx sxy) / denom
// println["got here, denom=$denom"]
// println["N=$N, sxy=$sxy, sx=$sx, sy=$sy"]
// TODO: make these arbitrary-precision sqrt? That means we have to
// fix arbitrary-precision sqrt so it takes units of measure.
r = (N sxy - sx sy) / (sqrt[denom] sqrt[N sy2 - sy^2])
// println["Got past r"]
return [a, b, r]
}
/** This performs a linear fit of the specified data as above, but instead
of returning the coefficients, it returns a function representing the
line, or, in other words, an anonymous function f that represents
y = f[x] of the best-fit line, or in more Frink-like notation, the
anonymous function looks like:
{|x| a x + b }
*/
linearFitFunction[data] :=
{
[a, b, r] = linearFit[data]
return parseToExpression["{|x| (" + inputForm[a] + ") x + (" + inputForm[b] + ")}"]
}
/** This performs a linear fit of the specified data as above, but instead
of returning the coefficients, it returns an expression representing the
line with x as the unknown variable. For example,
a x + b
*/
linearFitExpression[data] :=
{
[a, b, r] = linearFit[data]
return parseToExpression["(" + inputForm[a] + ") x + (" + inputForm[b] + ")"]
}
/** Performs a best quadratic fit of the specified data points. In other words,
this finds the coefficients a,b,c of a quadratic equation, that is,
the equation:
y = a x^2 + b x + c
params:
data: an array or set of [x,y] pairs.
returns:
[a, b, c, r]
where
r is the correlation coefficient
*/
quadraticFit[data] :=
{
array = toArray[data]
N = length[array]
if N < 3
return [undef, undef, undef, undef]
// We do it this way to preserve units of measure by initializing the sum
// with the first element.
[x,y] = array@0
sx = x // Sum of x
sy = y // Sum of y
sxy = x y // Sum of x*y
sx2 = x^2 // Sum of x^2
sx3 = x^3 // Sum of x^3
sx4 = x^4 // Sum of x^4
sy2 = y^2 // Sum of y^2
sx2y = x^2 y // Sum of x^2 * y
for i = 1 to N-1
{
[x,y] = array@i
sxy = sxy + x y
sx = sx + x
sy = sy + y
sx2 = sx2 + x^2
sx3 = sx3 + x^3
sx4 = sx4 + x^4
sy2 = sy2 + y^2
sx2y = sx2y + x^2 y
}
// These symbol changes make it more concise and match the symbols
// in Jean Meeus, Astronomical Algorithms, 4.5 and 4.6
P = sx
Q = sx2
R = sx3
S = sx4
T = sy
U = sxy
V = sx2y
D = N Q S + 2 P Q R - Q^3 - P^2 S - N R^2
a = (N Q V + P R T + P Q U - Q^2 T - P^2 V - N R U) / D
b = (N S U + P Q V + Q R T - Q^2 U - P S T - N R V) / D
c = (Q S T + Q R U + P R V - Q^2 V - P S U - R^2 T) / D
meany = sy / N
[x,y] = array@0
SSE = (y - a x^2 - b x - c)^2
SST = (y - meany)^2
for i = 1 to N-1
{
[x,y] = array@i
SSE = SSE + (y - a x^2 - b x - c)^2
SST = SST + (y - meany)^2
}
r = sqrt[1 - SSE/SST]
return [a, b, c, r]
}
/** This performs a quadratic fit of the specified data as above, but instead
of returning the coefficients, it returns an expression representing the
fit with x as the unknown. For example:
a x^2 + b x + c
*/
quadraticFitExpression[data] :=
{
[a, b, c, r] = quadraticFit[data]
return parseToExpression["(" + inputForm[a] + ") x^2 + (" + inputForm[b] + ") x + (" + inputForm[c] + ")"]
}
/** This performs a quadratic fit of the specified data as above, but instead
of returning the coefficients, it returns a function representing the
function, or, in other words, an anonymous function f that represents
y = f[x] of the best-fit line, or in more Frink-like notation, the
anonymous function looks like:
{|x| a x^2 + b x + c }
*/
quadraticFitFunction[data] :=
{
[a, b, c, r] = quadraticFit[data]
return parseToExpression["{|x| (" + inputForm[a] + ") x^2 + (" + inputForm[b] + ") x + (" + inputForm[c] + ")}"]
}
/** Fit an exponential function of the form y = A e^(B x).
See:
https://mathworld.wolfram.com/LeastSquaresFittingExponential.html
This uses the former fit on the page. It gives more weight to small y
values.
It was not originally clear how to make these equations have units of
measure. Taking, say, the logarithm or the exp of anything but a
dimensionless term makes no sense. The product (B x) must be
dimensionless (to be able to take the exponential function) so the
dimensions of B must be the dimensions of (1/x). Similarly, A and y must
have the same dimensions. But to take ln[y] means that y *must* be
dimensionless (which is unnecessary. It could have the same dimensions as
A.)
returns:
[A, B]
*/
exponentialFit[data] :=
{
[xa, ya] = unzip[data]
len = length[xa]
[x, xdim] = factorUnits[xa@0]
[y, ydim] = factorUnits[ya@0]
lny = ln[y]
a1n = 0 y
a2n = 0 x^2
a3n = 0 x
a4n = 0 x lny
for i=0 to len-1
{
x = xa@i / xdim
y = ya@i / ydim
lny = ln[y]
a1n = a1n + lny
a2n = a2n + x^2
a3n = a3n + x
a4n = a4n + x lny
}
denom = len a2n - a3n^2
a = (a1n a2n - a3n a4n) / denom
b = (len a4n - a3n a1n) / denom
return [exp[a] ydim, b / xdim]
}
/** This performs an exponential fit of the specified data as above, but
instead of returning the coefficients, it returns an expression
representing the fit with x as the unknown.
A e^(B x)
*/
exponentialFitExpression[data] :=
{
[A, B] = exponentialFit[data]
return parseToExpression["(" + inputForm[A] + ") e^((" + inputForm[B] + ")*(" + inputForm[x] + "))"]
}
/** This performs an exponential fit of the specified data as above, but
instead of returning the coefficients, it returns a function representing
the function, or, in other words, an anonymous function f that represents
y = f[x] of the best-fit line, or in more Frink-like notation, the
anonymous function looks like:
{|x| A e^(B x)}
*/
exponentialFitFunction[data] :=
{
[A, B] = exponentialFit[data]
return parseToExpression["{|x| (" + inputForm[A] + ") e^((" + inputForm[B] + ")*(" + inputForm[x] + "))}"]
}
/** Fit an exponential function of the form y = A e^(B x).
See:
https://mathworld.wolfram.com/LeastSquaresFittingExponential.html
This uses the latter fit on the page. It gives more weight to higher y
values.
It was not originally clear how to make these equations have units of
measure. Taking, say, the logarithm or the exp of anything but a
dimensionless term makes no sense. The product (B x) must be
dimensionless (to be able to take the exponential function) so the
dimensions of B must be the dimensions of (1/x). Similarly, A and y must
have the same dimensions. But to take ln[y] means that y *must* be
dimensionless (which is unnecessary. It could have the same dimensions as
A.)
returns:
[A, B]
*/
exponentialFit1[data] :=
{
[xa, ya] = unzip[data]
len = length[xa]
[x, xdim] = factorUnits[xa@0]
[y, ydim] = factorUnits[ya@0]
lny = ln[y]
a1n = 0 x^2y
a2n = 0 y
a3n = 0 x y
a4n = 0 x y lny
a1d = 0 y
for i=0 to len-1
{
x = xa@i / xdim
y = ya@i / ydim
lny = ln[y]
a1n = a1n + x^2 y
a2n = a2n + y lny
a3n = a3n + x y
a4n = a4n + x y lny
a1d = a1d + y
}
denom = (a1d a1n - a3n^2)
a = (a1n a2n - a3n a4n) / denom
b = (a1d a4n - a3n a2n) / denom
return [exp[a] ydim, b / xdim]
}
/** This performs an exponential fit of the specified data as above, but
instead of returning the coefficients, it returns an expression
representing the fit with x as the unknown.
A e^(B x)
*/
exponentialFitExpression1[data] :=
{
[A, B] = exponentialFit1[data]
return parseToExpression["(" + inputForm[A] + ") e^((" + inputForm[B] + ")*(" + inputForm[x] + "))"]
}
/** This performs an exponential fit of the specified data as above, but
instead of returning the coefficients, it returns a function representing
the function, or, in other words, an anonymous function f that represents
y = f[x] of the best-fit line, or in more Frink-like notation, the
anonymous function looks like:
{|x| A e^(B x)}
*/
exponentialFitFunction1[data] :=
{
[A, B] = exponentialFit1[data]
return parseToExpression["{|x| (" + inputForm[A] + ") e^((" + inputForm[B] + ")*(" + inputForm[x] + "))}"]
}
/** Calculates the r-value of the correlation between the yvalues and the fit
function's predicted yvalues. See
https://stats.libretexts.org/Bookshelves/Introductory_Statistics/OpenIntro_Statistics_(Diez_et_al)./07%3A_Introduction_to_Linear_Regression/7.02%3A_Line_Fitting_Residuals_and_Correlation */
rValue[function, data] :=
{
[xvalues, yvalues] = unzip[data]
ycalc = map[function, xvalues]
len = length[yvalues]
[meanx, sdx] = meanAndSD[yvalues, true]
[meany, sdy] = meanAndSD[ycalc, true]
//println["mean x: $meanx sdx: $sdx"]
//println["mean y: $meany sdy: $sdy"]
sum = undef // Make units come out right
for i=rangeOf[yvalues]
{
term = ((yvalues@i - meanx) / sdx) * ((ycalc@i - meany) / sdy)
if sum == undef
sum = term
else
sum = sum + term
}
return sum / (len-1)
}
/** Attempts to optimize an exponential fit by breaking it into 2 piecewise
segments each with different coefficients.
returns [function, rValue]
*/
exponentialFitFunctionPiecewise[data, debug=false] :=
{
sorted = sort[deepCopy[data], byColumn[0]]
len = length[sorted]
bestR = -1000
bestF = undef
for cutBefore = 3 to len-3
{
cutx = sorted@cutBefore@0 // x value after cut point
data1 = first[sorted, cutBefore]
data2 = rest[sorted, cutBefore]
e1 = exponentialFitExpression[data1]
e2 = exponentialFitExpression[data2]
fs = "{|x| \n" +
" if x < " + inputForm[cutx] + "\n" +
" return " + inputForm[e1] + "\n" +
" else\n" +
" return " + inputForm[e2] + "\n" +
"}"
if debug
println[fs]
f = parseToExpression[fs]
r = rValue[f, data]
if debug
print["r = $r "]
if r > bestR
{
bestR = r
bestF = f
if debug
println["*"]
}
if debug
println["\n"]
}
return [bestF, bestR]
}
/** Attempts to optimize a quadratic fit by breaking it into 2 piecewise
segments each with different coefficients.
returns [function, rValue]
*/
quadraticFitFunctionPiecewise[data, debug=false] :=
{
sorted = sort[deepCopy[data], byColumn[0]]
len = length[sorted]
bestR = -1000
bestF = undef
for cutBefore = 3 to len-3
{
cutx = sorted@cutBefore@0 // x value after cut point
data1 = first[sorted, cutBefore]
data2 = rest[sorted, cutBefore]
e1 = quadraticFitExpression[data1]
e2 = quadraticFitExpression[data2]
fs = "{|x| \n" +
" if x < " + inputForm[cutx] + "\n" +
" return " + inputForm[e1] + "\n" +
" else\n" +
" return " + inputForm[e2] + "\n" +
"}"
if debug
println[fs]
f = parseToExpression[fs]
r = rValue[f, data]
if debug
print["r = $r "]
if r > bestR
{
bestR = r
bestF = f
if debug
println["*"]
}
if debug
println["\n"]
}
return [bestF, bestR]
}
/** Attempts to optimize a quadratic fit by breaking it into 3 piecewise
segments each with different coefficients.
returns [function, rValue]
*/
quadraticFitFunctionPiecewise3[data, debug=false] :=
{
sorted = sort[deepCopy[data], byColumn[0]]
len = length[sorted]
bestR = -1000
bestF = undef
for cutBefore1 = 3 to len-6
{
cutx1 = sorted@cutBefore1@0 // x value after cut point1
data1 = slice[sorted, 0, cutBefore1]
e1 = quadraticFitExpression[data1]
for cutBefore2 = cutBefore1 + 3 to len-3
{
data2 = slice[sorted, cutBefore1, cutBefore2]
data3 = slice[sorted, cutBefore2, len]
cutx2 = data3@0@0 // x value after cut point 2
e2 = quadraticFitExpression[data2]
e3 = quadraticFitExpression[data3]
fs = "{|x| \n" +
" if x < " + inputForm[cutx1] + "\n" +
" return " + inputForm[e1] + "\n" +
" else\n" +
" if x < " + inputForm[cutx2] + "\n" +
" return " + inputForm[e2] + "\n" +
" else\n" +
" return " + inputForm[e3] + "\n" +
"}"
if debug
println[fs]
f = parseToExpression[fs]
r = rValue[f, data]
if debug
print["r = $r "]
if r > bestR
{
bestR = r
bestF = f
if debug
println["*"]
}
if debug
println["\n"]
}
}
return [bestF, bestR]
}
Download or view curveFit.frink in plain text format
This is a program written in the programming language Frink.
For more information, view the Frink
Documentation or see More Sample Frink Programs.
Alan Eliasen was born 20145 days, 13 hours, 35 minutes ago.