… or other irregular cyclic data.

Another Eng-Tips question asked how to approximate ice area over a freeze-thaw cycle using a function based on a sine or cosine curve. The screen-shot below shows three alternatives:

Using built in Excel functions requires a separate function for the freeze and thaw part of the cycle:

#### =(IF(DOY<DOY_FS,-1,IF(DOY>DOY_FE,1,SIN((DOY-F_Mid)/F_Days*PI())))+1)/2*A_max

=(IF(DOY<DOY_MS,1,IF(DOY>DOY_ME,-1,COS((DOY-DOY_MS)/M_Days*PI())))+1)/2*A_max

I have incorporated these in a short user defined function (UDF), which returns a column array of the full data range. (See Using Array Functions and UDFs if you are not familiar with array functions).

Function ScaleSin(DatRange As Variant, Outx As Variant) Dim Inc0x As Double, Inc0y As Double, Inc1x As Double, Inc1y As Double Dim Dec0x As Double, Dec0y As Double, Dec1x As Double, Dec1y As Double Dim NumX As Long, i As Long, ResA() As Double, OutXA() As Double, DX As Double, DY As Double Dim Pi As Double Pi = Atn(1) * 4 DatRange = DatRange.Value2 Inc0x = DatRange(1, 1) Inc0y = DatRange(1, 2) Inc1x = DatRange(2, 1) Inc1y = DatRange(2, 2) Dec0x = DatRange(3, 1) Dec0y = DatRange(3, 2) Dec1x = DatRange(4, 1) Dec1y = DatRange(4, 2) Outx = Outx.Value2 NumX = UBound(Outx) ReDim ResA(1 To NumX, 1 To 1) ReDim OutXA(1 To NumX, 1 To 1) i = 1 Do While i <= NumX Do While Outx(i, 1) < Inc0x ResA(i, 1) = Inc0y i = i + 1 Loop DX = Inc1x - Inc0x DY = Inc1y - Inc0y Do While Outx(i, 1) < Inc1x OutXA(i, 1) = (Outx(i, 1) - Inc0x) / DX * Pi - Pi / 2 ResA(i, 1) = Inc0y + DY * (Sin(OutXA(i, 1)) + 1) / 2 i = i + 1 Loop DX = Dec0x - Inc1x DY = Dec0y - Inc1y Do While Outx(i, 1) < Dec0x ResA(i, 1) = Inc1y + DY * (Outx(i, 1) - Inc1x) / DX i = i + 1 Loop DX = Dec1x - Dec0x DY = Dec0y - Dec1y Do While Outx(i, 1) < Dec1x OutXA(i, 1) = (Outx(i, 1) - Dec0x) / DX * Pi + Pi / 2 ResA(i, 1) = Dec1y + DY * (Sin(OutXA(i, 1)) + 1) / 2 i = i + 1 Loop Do While i <= NumX ResA(i, 1) = Dec1y i = i + 1 Loop i = i + 1 Loop ScaleSin = ResA End Function

An alternative approach suggested at the Eng-Tips discussion is to use a Sigmoid function of the form:

I have written another UDF to return such a function:

Function Sigmoid(xA As Variant, Optional a As Double = 1, Optional b As Double = 1, Optional c As Double = 1, _ Optional d As Double = 1, Optional f As Double = -5, Optional t As Double = 0) Dim Z As Double, NumX As Long, x As Double, i As Long, ResA() As Double xA = xA.Value2 If IsArray(xA) Then NumX = UBound(xA) Else NumX = 1 End If ReDim ResA(1 To NumX, 1 To 1) For i = 1 To NumX If NumX = 1 Then x = d * (xA - f) Else x = d * (xA(i, 1) - f) End If If x >= 0 Then ResA(i, 1) = a / (b + c * Exp(-x)) + t Else Z = Exp(x) ResA(i, 1) = a * Z / (b + c * Z) + t End If Next i Sigmoid = ResA End Function

### The sigmoid function also returns an array, but must be entered separately for the freeze and thaw part of the cycle.

### Examples of the use of both functions applied to freeze-thaw data are given in the download file: Freeze-sin.xlsb

As usual, the download file includes full open-source code.