Download or view LeastSquares.frink in plain text format
use Matrix.frink
/** This class finds least-squares curve fits. It takes a set of points in x
and y and returns the best fit for a given curve type.
It allows you to specify the "basis functions" (or basis expressions) for
the curve type. For example, if you wanted to find the best linear fit,
the basis functions would be [x, 1]. For a quadratic (squared) fit, the
basis functions would be [x^2, x, 1].
This class finds the coefficients that best fit the provided points. That
is, for the fit to a line mentioned above, this would calculate the
coefficients c1, c2 to best solve
y = c1 x + c2
This uses the Matrix.frink class to perform its solution, notably the
leastSquares[] method.
See the leastSquaresTest.frink program for examples of use of this library.
Curve-fitting can be performed on an overdetermined system, where there are
more measurements than equations.
This is the best discussion I've seen of least-squares fitting:
https://www.aleksandrhovhannisyan.com/blog/the-method-of-least-squares/
https://www.aleksandrhovhannisyan.com/blog/least-squares-fitting/
See:
https://mathworld.wolfram.com/LeastSquaresFitting.html
See also for special curve fits:
https://mathworld.wolfram.com/LeastSquaresFittingExponential.html
https://mathworld.wolfram.com/LeastSquaresFittingLogarithmic.html
https://mathworld.wolfram.com/LeastSquaresFittingPowerLaw.html
Also see curveFit.frink for clear examples of linear and quadratic fit.
*/
class LeastSquares
{
/** An array of x values to fit. */
var xvalues
/** An array of y values to fit. */
var yvalues
/** An array of basis expressions. */
var basisExprs
/** The variable which contains the curve fit coefficients. It is a
one-column matrix */
var sol
/** Construct and solve the system. */
new[xvalues, yvalues, basisExprs] :=
{
this.xvalues = xvalues
this.yvalues = yvalues
this.basisExprs = basisExprs
sol = fit[]
}
class fitLinear[xvalues, yvalues] :=
{
return fitDegree[xvalues, yvalues, 1]
}
class fitQuadratic[xvalues, yvalues] :=
{
return fitDegree[xvalues, yvalues, 2]
}
class fitCubic[xvalues, yvalues] :=
{
return fitDegree[xvalues, yvalues, 3]
}
class fitDegree[xvalues, yvalues, degree] :=
{
// Make basis functions like [x^2, x, 1]
a = new array
for i = 0 to degree
a.pushFirst[constructExpression["Power", [noEval[x], i]]]
return new LeastSquares[xvalues, yvalues, a]
}
/** Performs the internal curve fitting. Solutions are placed into the
variable sol which is a one-column Matrix. */
fit[] :=
{
rows = length[xvalues]
cols = length[basisExprs]
a = new array[[rows,cols],0]
for row = 0 to rows-1
for col = 0 to cols-1
{
x = xvalues@row
a@row@col = eval[basisExprs@col]
}
A = new Matrix[a]
B = new Matrix[yvalues.transpose[]]
return A.leastSquares[B]
}
/** Returns an string representing the best fit. For example, this
might return
"3.21 x + 1.54"
for a linear fit.
*/
toExpressionString[varname = "x"] :=
{
cols = length[basisExprs]
estr = ""
for col = 0 to cols-1
{
be2 = substituteExpression[basisExprs@col, parseToExpression["x"], parseToExpression[varname]]
estr = estr + "(" + inputForm[sol.get[col+1, 1]] + " * " + inputForm[be2] + ")"
if col < cols-1
estr = estr + " + "
}
return estr
}
/** Returns an expression representing the best fit. For example, this
might return
3.21 x + 1.54
for a linear fit.
*/
toExpression[varname = "x"] :=
{
return parseToExpression[toExpressionString[varname]]
}
/** Returns an anonymous single-argument function that represents the best
fit, which you can then use to calculate additional y values. For
example, you could call it like:
f = toFunction[]
y1 = f[1]
y2 = f[2]
*/
toFunction[varname = "x"] :=
{
varsym = parseToExpression[varname]
return constructExpression["AnonymousFunction", [[varsym], toExpression[varname]]]
}
/** Returns the solution coefficients as a 1-column Matrix. */
toMatrix[] :=
{
return sol
}
/** Returns the solution coefficients as a row array. */
toArray[] :=
{
return sol.getColumnAsArray[1]
}
/** Calculate the RMS of the residuals. */
residual[] :=
{
f = toFunction[]
size = length[xvalues]
r = 0 (yvalues@0)^2 // Make units work out
for i = 0 to size-1
r = r + (yvalues@i - f[xvalues@i])^2
return sqrt[r]
}
/** 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[] :=
{
f = toFunction[]
ycalc = map[f, 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)
}
/** Calculates the mean and standard deviation of an array or
enumerating expression.
This uses Welford's algorithm as cited in Knuth, The Art of Computer
Programming, Vol. 2, 3rd edition, page 232.
Arguments:
[list, sample]
where
list: is an array or enumerating expression of the items to average.
sample: a boolean flag indicating if we want the standard deviation to be
the sample standard variation (=true) or the population standard
variation (=false).
Returns:
[mean, sd, number]
*/
class meanAndSD[list, sample] :=
{
M = 0 list@0 // Make units of measure come out right
S = 0 (list@0)^2 // Make units of measure come out right
k = 1
for v = list
{
oldM = M
diff = v-oldM
M = M + diff / k
S = S + diff * (v - M)
k = k+1
}
if sample == true
sub = 1 // Sample standard deviation, subtract 1 from num
else
sub = 0 // Population standard deviation
return [M, sqrt[S/(k-1-sub)], k-1]
}
}
Download or view LeastSquares.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 19966 days, 1 hours, 15 minutes ago.