Test Smoothing_F Class about fda HOT 8 OPEN

@DongYuYu I'd like for you to work on this issue #5. Before you can properly test the Smoothing_F class, you'll need to determine the cause of the zero-column in the fit produced by RidgeRegression during the training phase of Smoothing_F. You can use this issue to discuss this issue with the group.

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Quick Summary for Friday Work

I think that's all we done last Friday as following:

B_Spline:

1. Add bs for shifting the j-th spline to j+mMax-m-th spline evaluation

2. define range(m:Int) = 0 to ττ.dim - 1+m -2 to modify the index range of spline

3. decrease the augment interval for τ from mMax to mMax-1

4. change the boarder value for τ to ττ(0)-0,01 and ττ(ττ.dim - 1)+0.01

5. check if the last spline works well for the setting above

Smoothing_F:

1. change the function usage at predict and form_phi from bb to bs

2. change the range for makeknots by multiply *t(t.dim-1)

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@DongYuYu I've cleaned up the B_Spline class in commit b829109 based on your edits.

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@DongYuYu Any progress on this? I'm particularly interested in the testing of RidgeRegression with the penalty matrix.

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Not much yet. I'm into RedgeRegression and might ask questions later.

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@mepcotterell I have implemented some part of the regression. I added the penalty matrix in

RidgeRegression:
    def xtx(Σ:MatrixD): MatrixD = {
        val a = x.t * x
        for (i <- 0 until a.dim1) {
            for (j <- 0 until a.dim2) {
                a(i, j) += lambda*Σ(i,j)
            }
        }
        a
    }

However, I have some problems: First, regarding to the this formula in R,
(X' W X + λ Σ) c = X' W y, I think in our regression formula, we take the X.t but here is it taking derivative? We will see.
Secondly, I think Σ suppose to be a square matrix here, but Σ[i,j] = Integral B''[i](t) B''[j](t) dt, is i necessary to be equal to j here? I still not very clear about the B-Spline formulation process.

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@DongYuYu It's unfortunate that they mix/overload notations, but I believe X' denotes the transpose of the matrix X and B'' denotes the second derivative of the function B.

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@DongYuYu See B_SplineTest5 in B_Spline for an implementation of the Σ matrix.

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