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|
unit unitRegressionPolynomiale;
// Régression polynomiale d'un ensemble de points XY
// On n'utilise aucune dépendance (GHTopo, etc ...)
{$mode Delphi}
interface
uses
Classes, SysUtils, Math;
type
{ TRPPointXY }
TRPPointXY = record
X : double;
YBrut : double;
YFitted: double;
procedure setFrom(const QX, QY: double);
procedure setYFitted(const Y: double);
end;
type
{ TRPArrayOfFloats }
TRPArrayOfFloats = record
private
FIdxFirstNotNull: integer;
FIdxLastNotNull : integer;
FArray: array of double;
public
property IdxFirstNotNull: integer read FIdxFirstNotNull;
property IdxLastNotNull: integer read FIdxLastNotNull;
procedure RedimWoInit(n: integer);
procedure ReDim(const N: integer; const V, V0: double); // Redim initialise avec des zéros
procedure AppendFrom(const V: double);
procedure AddValueTo(const Idx: integer; const V: double);
procedure setFrom(const Idx: integer; const QValue: double);
function getFrom(const Idx: integer): double;
function getLast(): double;
function getMinValue(): double;
function getMaxValue(): double;
function getNbElements(): integer;
procedure Exchange(const k, m: integer);
function toString(const Sep: string): string;
function setFromStr(const MyStr: string; const Sep: string): boolean;
end;
type
{ TRPMatrix }
TRPMatrix = record
private
FRows: array of TRPArrayOfFloats;
procedure ExchangeCol(const k, m: integer);
public
procedure Redim(const NbRows, NbCols: integer);
function GetValue(const R, C: integer): double;
procedure SetValue(const R, C: integer; const V: double);
function getNbRows(): integer;
function getNbCols(): integer;
function SetRowFromStr(const aRow: integer; const MyStr: string; const Sep: string): boolean;
function IsZero(const R, C: integer): boolean;
function Determinant(): double;
function IsSquared(): boolean;
function RowToStr(const r: integer; const Sep: string): string;
function SwapColonneAndVecteur(var B: TRPArrayOfFloats; const IdxColonne: integer): boolean;
procedure SwapRows(const i, j: integer);
end;
type
{ TRegressionPolynomiale }
TRegressionPolynomiale = class(TList)
private
FCoeffsPoly: TRPArrayOfFloats;
FMatrixC : TRPMatrix;
FNbRejectedValues : integer;
public
property NbRejectedValues: integer read FNbRejectedValues;
property MatrixC: TRPMatrix read FMatrixC;
property CoefPolynome: TRPArrayOfFloats read FCoeffsPoly;
function Initialiser(): boolean;
procedure ClearListe();
procedure Finaliser();
function GetNbElements(): integer;
procedure AddElement(const E: TRPPointXY); overload;
procedure AddElement(const QX, QY: double); overload;
function GetElement(const Idx: integer): TRPPointXY;
procedure SetElement(const Idx: integer; const E: TRPPointXY);
procedure InsertElement(const Idx: integer; const E: TRPPointXY);
procedure SortByXValues();
function CalcInterpolatedValueY(const X: double): double;
function Regression(const DegreOfPolynom: integer; const QValeurMini: double): boolean;
function ExportPolynomeAsOOoFormula(const CellName: string): string;
end;
implementation
// Affichage des messages d'erreur. Cà va du simple WriteLn() à une procédure plus complexe à implémenter
procedure AfficherMessageErreur(const Msg: string);
begin
WriteLn(Msg);
end;
procedure QSwap(var V1, V2: double); overload;
var Tmp: double;
begin
Tmp := V1;
V1 := V2;
V2 := Tmp;
end;
// Trier les visées par TagDouble (e.g. distance)
function SortByX(Item1, Item2: Pointer): Integer;
var
E1, E2: ^Double;
bidon1: double;
bidon2: double;
begin
E1 := Item1;
E2 := Item2;
bidon1 := E1^;
bidon2 := E2^;
if (bidon1 < bidon2) then Result := -1
else if (bidon1 = bidon2) then Result := 0
else Result := 1;
end;
{ TRPMatrix }
procedure TRPMatrix.Redim(const NbRows, NbCols: integer);
var
i: Integer;
begin
SetLength(FRows, NbRows); //, NbCols);
for i := 0 to High(FRows) do
begin
FRows[i].ReDim(NbCols, 0.0, 0.0);
end;
end;
function TRPMatrix.GetValue(const R, C: integer): double;
begin
Result := FRows[R].getFrom(C);
end;
procedure TRPMatrix.SetValue(const R, C: integer; const V: double);
begin
FRows[R].setFrom(C, V);
end;
function TRPMatrix.getNbRows: integer;
begin
result := length(FRows);
end;
function TRPMatrix.getNbCols: integer;
begin
result := FRows[0].getNbElements();
end;
function TRPMatrix.SetRowFromStr(const aRow: integer; const MyStr: string; const Sep: string): boolean;
begin
result := FRows[aRow].setFromStr(MyStr, sep);
end;
function TRPMatrix.IsZero(const R, C: integer): boolean;
var
Z: Double;
begin
Z := FRows[R].getFrom(C);
result := (Math.IsZero(Z, 1E-12));
end;
procedure TRPMatrix.ExchangeCol(const k, m: integer);
var
q: Integer;
begin
{ On échange la colonne k et la colonne m }
for q := 0 to High(FRows) do FRows[q].Exchange(k, m);
end;
function TRPMatrix.Determinant(): double;
var
k, m, q, r : Integer;
s , t: Double;
n: SizeInt;
begin
n := Length(FRows) - 1;
for k := 0 to n do
begin
m := k;
while (self.IsZero(k, m)) and (m < n) do inc(m); //while (IsZero(c2[k, m], 1E-9)) and (m < FDegrePolyn) do inc(m);
if (self.IsZero(n, k)) then Exit(0)
else
begin
{ On échange la colonne k et la colonne m }
self.ExchangeCol(k, m);
{ Pour les colonnes suivantes : Cq <- Cq - (aq / am) * Cm }
for q := k + 1 to n do
begin
s := self.GetValue(k, q) / self.GetValue(k, k);
for r := 0 to n do self.setValue(r, q, self.GetValue(r, q) - s * self.GetValue(r, k));
end;
end;
end;
Result := 1;
for k := 0 to n do Result := Result * self.GetValue(k, k);
AfficherMessageErreur('Déterminant de la matrice: %.15f', [result]);
end;
function TRPMatrix.IsSquared: boolean;
var
m, n: Integer;
begin
m := self.getNbRows();
n := self.getNbCols();
Result := (n > 1) AND (m > 1) AND (m = n);
end;
function TRPMatrix.RowToStr(const r: integer; const Sep: string): string;
var
n, c: Integer;
begin
n := getNbCols();
Result := format('Row %d of %d: ', [r, n-1]);
for c := 0 to n-1 do Result += format('%.3f%s', [self.GetValue(r, c), Sep]);
end;
function TRPMatrix.SwapColonneAndVecteur(var B: TRPArrayOfFloats; const IdxColonne: integer): boolean;
var
r, n: Integer;
Mij, BB: Double;
begin
result := false;
n := B.getNbElements();
if (n < self.getNbRows()) then exit(false);
for r := 0 to n-1 do
begin
Mij := self.GetValue(r, IdxColonne);
BB := B.getFrom(r);
QSwap(Mij, BB);
self.setValue(r, IdxColonne, Mij);
B.setFrom(r, BB);
end;
result := true;
end;
procedure TRPMatrix.SwapRows(const i, j: integer);
var
tmp: TRPArrayOfFloats;
begin
tmp := self.FRows[i];
self.FRows[i] := self.FRows[j];
self.FRows[j] := tmp;
end;
{ TRegressionPolynomiale }
function TRegressionPolynomiale.Initialiser(): boolean;
begin
result := false;
FNbRejectedValues := 0;
ClearListe();
result := true;
end;
procedure TRegressionPolynomiale.ClearListe();
var
n, i: Integer;
begin
FNbRejectedValues := 0;
FMatrixC.Redim(0,0);
FCoeffsPoly.ReDim(0, 0.00, 0.00);
n := self.Count;
if (n > 0) then
begin
for i := Count-1 downto 0 Do
begin
if (self.Items[i] <> Nil) then Dispose(self.Items[i]); // Libération
self.Delete(i); // Suppression de l'élément
end;
self.clear;
end;
end;
procedure TRegressionPolynomiale.Finaliser();
begin
self.ClearListe();
end;
function TRegressionPolynomiale.GetNbElements: integer;
begin
Result := self.Count;
end;
procedure TRegressionPolynomiale.AddElement(const E: TRPPointXY);
var
pE: ^TRPPointXY;
begin
New(pE);
pE^ := E;
self.Add(pE);
end;
procedure TRegressionPolynomiale.AddElement(const QX, QY: double);
var
P: TRPPointXY;
begin
P.setFrom(QX, QY);
self.AddElement(P);
(*
if (QY >= QYMini) then
begin
P.setFrom(QX, QY);
self.AddElement(P);
end
else
FNbRejectedValues += 1;
//*)
end;
function TRegressionPolynomiale.GetElement(const Idx: integer): TRPPointXY;
begin
Result := TRPPointXY(Items[Idx]^);
end;
procedure TRegressionPolynomiale.InsertElement(const Idx: integer; const E: TRPPointXY);
var
pE: ^TRPPointXY;
begin
New(pE);
pE^ := E;
self.Insert(Idx, pE);
end;
procedure TRegressionPolynomiale.SortByXValues;
begin
self.Sort(SortByX());
end;
function TRegressionPolynomiale.CalcInterpolatedValueY(const X: double): double;
var
j: Integer;
begin
Result := 0.00;
for j := 0 to FCoeffsPoly.getNbElements() - 1 do Result += FCoeffsPoly.getFrom(j) * Power(X, j);
end;
function TRegressionPolynomiale.Regression(const DegreOfPolynom: integer; const QValeurMini: double): boolean;
var
FVecteurB: TRPArrayOfFloats;
i, j, k, Nb: Integer;
D, P: Double;
BB: double;
V: TRPPointXY;
procedure DispMatrix(const M: TRPMatrix; const Tag: string);
var
EWE: String;
l, c: Integer;
begin
AfficherMessageErreur('Matrice "%s" de taille %dx%d)', [Tag, M.getNbRows, M.getNbCols()]);
for l := 0 to M.getNbRows() - 1 do
begin
EWE := format('Row %d: ', [l]);
for c := 0 to M.getNbCols() - 1 do
EWE += format('%8.4f, ', [M.GetValue(l, c)]);
EWE += #13#10;
AfficherMessageErreur(EWE);
end;
AfficherMessageErreur('');
end;
function Solve(const MA: TRPMatrix; const VB: TRPArrayOfFloats; out VX: TRPArrayOfFloats): boolean;
var
QNb, jj, ii, kk: Integer;
Pivot, aaa, temp: Double;
MatriceAugmentee: TRPMatrix;
begin
result := false;
AfficherMessageErreur('Solve');
QNb := VB.getNbElements();
// construction de la matrice augmentée = on y met le vecteur second membre
MatriceAugmentee.Redim(QNb, QNb+1);
for ii := 00 to QNb-1 do
begin
for jj := 0 to QNb -1 do MatriceAugmentee.SetValue(ii, jj, MA.GetValue(ii, jj));
MatriceAugmentee.SetValue(ii, QNb, VB.getFrom(ii));
end;
for ii := 0 to QNb - 1 do
begin
AfficherMessageErreur('Row %d', [ii]);
Pivot := MatriceAugmentee.GetValue(ii,ii);
if (IsZero(Pivot)) then exit(false); // Pivot nul
// Normalisation de la ligne de la matrice A
for jj := 0 to QNb do
begin
aaa := MatriceAugmentee.GetValue(ii, jj);
MatriceAugmentee.SetValue(ii, jj, aaa / Pivot);
end;
// Elimination des autres lignes
for kk := 0 to QNb - 1 do
begin
if (kk <> ii) then
begin
temp := MatriceAugmentee.GetValue(kk, ii);
for jj := 0 to QNb do
begin
aaa := MatriceAugmentee.GetValue(kk, jj) - temp * MatriceAugmentee.GetValue(ii, jj);
MatriceAugmentee.SetValue(kk, jj, aaa);
end;
end;
end;
end; // for ii := 0 to QNb - 1 do
// Solution
VX.ReDim(QNb, 0.00, 0.00);
for ii := 0 to QNb-1 do VX.setFrom(ii, MatriceAugmentee.GetValue(ii, MatriceAugmentee.getNbCols() - 1));
// purges
MatriceAugmentee.Redim(0,0);
// OK ?
result := True;
// Code C original:
(*
void pivotGauss(double A[MAX_SIZE][MAX_SIZE + 1], int N) {
for (int i = 0; i < N; i++)
{
double pivot = A[i][i];
if (pivot == 0.0) {
printf("Le pivot est nul. Le systeme n'a pas de solution unique.\n");
return;
}
// Normalisation de la ligne i
for (int j = 0; j < N + 1; j++) {
A[i][j] /= pivot;
}
// Elimination des autres lignes
for (int k = 0; k < N; k++) {
if (k != i) {
double temp = A[k][i];
for (int j = 0; j < N + 1; j++) {
A[k][j] -= temp * A[i][j];
}
}
}
}
}
//*)
end;
begin
result := false;
FCoeffsPoly.ReDim(DegreOfPolynom + 1, 0.00, 0.00);
Nb := self.GetNbElements();
AfficherMsgErrClass(self, 'Regression: IndiceMax: %d, DegPoly: %d', [Nb, DegreOfPolynom]);
FMatrixC.Redim(DegreOfPolynom + 1, DegreOfPolynom + 1);
FVecteurB.Redim(DegreOfPolynom + 1, 0.00, 0.00);
// construction de la matrice de régression
for i := 0 to DegreOfPolynom do
begin
for j := 0 to DegreOfPolynom do
begin
BB := 0.00;
for k := 0 to Nb-1 do
begin
V := self.GetElement(k);
if (V.YBrut > QValeurMini) then BB += IntPower(V.X, i + j);
end;
FMatrixC.SetValue(i, j, BB);
end;
end;
// et du Vecteur B
for i := 0 to DegreOfPolynom do
begin
BB := 0.00; // B[j, 0] := B[j, 0] + Points[i].Y * Power(Points[i].X, j);
for j := 0 to Nb-1 do
begin
V := self.GetElement(j);
if (V.YBrut > QValeurMini) then BB += V.YBrut * IntPower(V.X, i); // for j := 0 to IndiceMax do b[i] := b[i] + ArrValeursXY[j].YBrut * Power(ArrValeursXY[j].X, i);
end;
FVecteurB.setFrom(i, BB);
end;
//DispMatrix(FMatrixC, 'Matrice originale');
//AfficherMessageErreur('Vecteur B: %d éléments', [FVecteurB.getNbElements()]);
//for i := 0 to FVecteurB.getNbElements() - 1 do AfficherMessageErreur('%d: %.3f', [i, FVecteurB.getFrom(i)]);
//AfficherMessageErreur('');
if (Solve(FMatrixC, FVecteurB, FCoeffsPoly)) then
begin
AfficherMessageErreur(' -- Solution X: %d éléments', [FCoeffsPoly.getNbElements()]);
//for i := 0 to FCoeffsPoly.getNbElements() - 1 do
// AfficherMessageErreur('%d: %.3f', [i, FCoeffsPoly.getFrom(i)]);
//AfficherMessageErreur('');
AfficherMessageErreur(' -- Polynôme de régression de degré %d', [FCoeffsPoly.getNbElements() - 1]);
for i := 0 to FCoeffsPoly.getNbElements() - 1 do
begin
P := FCoeffsPoly.getFrom(i);
AfficherMessageErreur('%.8f X ** %d +', [P, i]);
end;
{ On calcule les valeurs interpolées}
AfficherMessageErreur(' -- Ajustement');
for i := 0 to Nb-1 do
begin
V := self.GetElement(i);
V.YFitted := CalcInterpolatedValueY(V.X);
self.SetElement(i, V);
end;
AfficherMessageErreur('');
// contrôle
AfficherMessageErreur('Valeurs ajustées');
for i := 0 to Nb-1 do
begin
V := self.GetElement(i);
AfficherMessageErreur(format('%d: %.3f, %.3f, %.3f: %.3f', [i, V.X, V.YBrut, V.YFitted, V.YFitted - V.YBrut]));
end;
end;
result := True;
end;
function TRegressionPolynomiale.ExportPolynomeAsOOoFormula(const CellName: string): string;
var
i, Nb: Integer;
begin
Result := '=';
Nb := FCoeffsPoly.getNbElements() - 1;
for i := 0 to Nb do
Result += Trim(format('(%s*%s^%d)%s', [FormatterNombreOOo(FCoeffsPoly.getFrom(i), 12), CellName, i, BoolToStr(i = Nb, '', '+')]));
end;
procedure TRegressionPolynomiale.SetElement(const Idx: integer; const E: TRPPointXY);
begin
try
if (Idx < 0) then exit;
TRPPointXY(Items[Idx]^) := E;
except
end;
end;
//******************************************************************************
{ TRPPointXY }
procedure TRPPointXY.setFrom(const QX, QY: double);
begin
self.X := QX;
self.YBrut := QY;
self.YFitted := self.YBrut;
end;
procedure TRPPointXY.setYFitted(const Y: double);
begin
self.YFitted := Y;
end;
{ TRPArrayOfFloats }
procedure TRPArrayOfFloats.RedimWoInit(n: integer);
begin
SetLength(self.FArray, n);
end;
procedure TRPArrayOfFloats.ReDim(const N: integer; const V, V0: double);
var
i: Integer;
begin
SetLength(FArray, N);
if (N > 0) then
begin
for i := Low(FArray) to High(FArray) do self.FArray[i] := V;
self.FArray[0] := V0;
end;
FIdxFirstNotNull := N-1;
FIdxLastNotNull := 0;
end;
procedure TRPArrayOfFloats.AppendFrom(const V: double);
var
Nb: Integer;
begin
Nb := length(FArray);
SetLength(FArray, Nb+1);
FArray[Nb] := V;
end;
procedure TRPArrayOfFloats.AddValueTo(const Idx: integer; const V: double);
begin
self.FArray[Idx] += V;
end;
procedure TRPArrayOfFloats.setFrom(const Idx: integer; const QValue: double);
begin
FArray[Idx] := QValue;
if (QValue <> 0.00) then
begin
FIdxFirstNotNull := Min(Idx, FIdxFirstNotNull);
FIdxLastNotNull := Max(Idx, FIdxLastNotNull);
end;
end;
function TRPArrayOfFloats.getFrom(const Idx: integer): double;
begin
Result := FArray[Idx];
end;
function TRPArrayOfFloats.getLast: double;
var
N: integer;
begin
N := length(FArray);
if (0 = N) then Result := 0.00 else Result := getFrom(N-1);
end;
function TRPArrayOfFloats.getMinValue(): double;
var
i: Integer;
begin
Result := Infinity;
for i := 0 to High(FArray) do result := min(Result, FArray[i]);
end;
function TRPArrayOfFloats.getMaxValue(): double;
var
i: Integer;
begin
Result := -Infinity;
for i := 0 to High(FArray) do result := max(Result, FArray[i]);
end;
function TRPArrayOfFloats.getNbElements(): integer;
begin
result := length(FArray);
end;
procedure TRPArrayOfFloats.Exchange(const k, m: integer);
var
V1, V2, Tmp: Double;
begin
V1 := FArray[k];
V2 := FArray[m];
QSwap(V1, V2);
FArray[k] := V1;
FArray[m] := V2;
end;
function TRPArrayOfFloats.toString(const Sep: string): string;
var
i: Integer;
begin
Result := '';
for i := 0 to High(FArray) do Result += format('%.3f%s', [self.getFrom(i), Sep]);
end;
function TRPArrayOfFloats.setFromStr(const MyStr: string; const Sep: string): boolean;
var
LSR: TStringList;
S1, S2 : string;
P, i: Integer;
begin
Result := false;
self.ReDim(0,0,0);
LSR := TStringList.Create;
try
S1 := '';
S2 := Trim(MyStr);
if (S2 = '') then exit(false);
try
if (not Assigned(LSR)) then raise Exception.Create('SplitToTStringList: Le paramètre LSR doit être créé par TStringList.Create');
LSR.Sorted := false;
LSR.Duplicates := dupAccept;
LSR.Clear;
repeat
P := Pos(Sep, S2);
s1 := trim(Copy(s2,1,P - 1));
s2 := Trim(Copy(s2, P + 1,Length(s2)));
if (Length(S1) > 0) then LSR.Add(s1);
until P =0;
if (Length(S2) > 0) then LSR.Add(s2);
Result := (LSR.Count > 0);
if (LSR.Count > 0) then
begin
self.Redim(LSR.Count, 0.0, 0.0);
for i := 0 to LSR.Count - 1 do self.setFrom(i, StrToFloatDef(Trim(LSR.Strings[i]), 0.00));
LSR.Clear;
end;
except
end;
finally
FreeandNil(LSR);
end;
end;
end. |
TRegressionPolynomiale : Petite classe légère et sans prétention (exercice) pour régressions polynomiales
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Il y a des jours où j'éprouve une haine profonde envers microsoft 
et Apple c'est pas mieux 
sont 
(if, fonctions vides...)
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