This is how John Piper describes the Trinity:
"In a nutshell (following Jonathan Edwards), I would describe the Trinity like this: The Father is God existing in the primal, unoriginated, most absolute manner. The Son is God eternally generated by the Father’s having a clear and distinct idea or image of himself, so much so that his image or reflection of himself is God—the Son. The Holy Spirit is God existing as the infinite Spirit of love and delight flowing eternally between the Son and the Father."
See whole article here.
I like this definition and I think it's not too much to say it can be gathered from scripture. It is not sheer speculation. We know the Father begat the Son and that the Son perfectly reflects him, and we know the Spirit "proceeds" from the Father and the Son.
The Self-Image-Relationship (SIR) picture is a picture. It doesn't explain everything, but it does fit with Piper's description, and Piper's description fits with scripture.
Lee said that God is Triune for dispensing into mankind. This unfortunately says that without man God's triune-ness has no purpose. This seems a flawed view.
God's triune-ness had meaning and purpose to himself before he ever created man. Piper's description, which I tried lamely to expound on, sums up well what it is.