Suppose I measure two quantities at regular intervals, and I have N
pairs of measurements (X[1], Y[1])...(X[N], Y[N]).  I would like to
find the correlation between the two but assign a linear "weight" to
each pair of measurements, such that (X[1], Y[1]) has a weight of 1/N,
(X[2], Y[2]) has a weight of 2/N, etc, and (X[N], Y[N]) has a weight
of N/N or 1.  The weights are intended to give more relevance to
recent measurements and less to old ones.

What equations would I use to calculate the effective M, B, and R
values for a time-weighted correlation between X and Y as described above?

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Clarification of Question by gw-ga on 09 Oct 2004 07:03 PDT

I expect the equations to be based on a linear least-squares regression.

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Clarification of Question by gw-ga on 11 Oct 2004 17:49 PDT

The code I wish to adapt (I wrote it some time ago) is as follows:

{
  This procedure performs linear least-squares regression.

  M = slope
  B = Y-intercept

  R = correlation coefficient
  Sqr(R) = coefficient of determination

  if Sqr(R) = 1, then the line fits perfectly
  if Sqr(R) = 0, then the line fits very poorly

  Sqr(R) = the fraction of the variation in y that can be explained by
variations in x
}
procedure LinearLeastSquaresRegression(const X: TDoubleDynArray;
                                       const Y: TDoubleDynArray;
                                       out M: Double;
                                       out B: Double;
                                       out R: Double);
var
  I: Integer;     // loop index
  SX: Double;     // sum of X values
  SY: Double;     // sum of Y values
  SSX: Double;    // sum of squares of X values
  SXY: Double;    // sum of products of X and Y values
  N: Integer;     // number of data points
  D: Double;      //
  YAv: Double;    // average Y value
  SELine: Double; // sum of errors in predicted Y values
  SEAv: Double;   //
begin
  N := Length(X);
  Assert(N >= 2);
  Assert(Length(Y) = N);

  SX := 0;
  SY := 0;
  SSX := 0;
  SXY := 0;

  for I := 0 to (N - 1) do
  begin
    SX := SX + X[I];
    SY := SY + Y[I];
    SSX := SSX + Sqr(X[I]);
    SXY := SXY + X[I] * Y[I];
  end;

  D := SSX * N - Sqr(SX);
  M := (SXY * N - SX * SY) / D;
  B := (SSX * SY - SX * SXY) / D;
  YAv := SY / N;

  SELine := 0;
  SEAv := 0;

  for I := 0 to (N - 1) do
  begin
    SELine := SELine + Sqr(Y[I] - (M * X[I] + B));
    SEAv := SEAv + Sqr(Y[I] - YAv);
  end;

  R := Sqrt(1 - SELine / SEAv);
end;

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Clarification of Question by gw-ga on 11 Oct 2004 18:05 PDT

Perhaps instead of using a weight from 0..1, it would be better if the
sum of all the weights was N.

This block of code might be the only thing I need to modify:

  for I := 0 to (N - 1) do
  begin
    SX := SX + X[I];
    SY := SY + Y[I];
    SXX := SXX + Sqr(X[I]);
    SXY := SXY + X[I] * Y[I];
  end;

Possible new code:

  for I := 0 to (N - 1) do
  begin
    W := 2 * (I + 0.5) / N; // weight for this time index

    SX := SX + W * X[I];
    SY := SY + W * Y[I];
    SXX := SXX + W * Sqr(X[I]);
    SXY := SXY + W * X[I] * Y[I];
  end;

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Clarification of Question by gw-ga on 11 Oct 2004 18:10 PDT

I guess that should have been

W := (I + 0.5) * 2 / N;

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Clarification of Question by gw-ga on 11 Oct 2004 18:13 PDT

Silly me.  Those are the same.