Chi-squared Distribution ($k>1$)

Theory

Given $k$ standard random variables $X_i \sim \mathcal{N}(0,1)$ for $i=1,\dots,k$ we would like to find the random variable $Y = \sum_{i=1}^k X_i^2$. The analytic solution is known: $Y$ follows a chi-squared distribution with $k$ degrees of freedom.

Using polynomial chaos expansion (PCE), the problem can be solved using Galerkin projection. Let $\{\phi_k \}_{k=0}^{n}$ be the monic orthogonal basis relative to the probability density of $X = [X_1, \dots, X_k]$, namely

\[f_X(x) = \prod_{i=1}^k \frac{1}{\sqrt{2 \pi}} \, \exp \left( - \frac{x_i^2}{2} \right).\]

Then, the PCE of $X_i$ is given by

\[X_i = \sum_{k=0}^n x_{i,k} \phi_k,\]

with

\[x_{i,0} = 0, \quad x_{i,i+1} = 1, \quad x_{i,l} = 0 \quad \forall l \in \{1,\dots,n\} \setminus \{i+1\}.\]

To find the PCE coefficients $y_k$ for $Y = \sum_{k=}^n y_k \phi_k$, we apply Galerkin projection, which leads to

\[y_m \langle \phi_m, \phi_m \rangle = \sum_{i=1}^k \sum_{j_1=0}^n \sum_{j_2=0}^n x_{i,j_1} x_{i,j_2} \langle \phi_{j_1} \phi_{j_2}, \phi_m \rangle \quad \forall m = 0, \dots, n.\]

Hence, knowing the scalars $\langle \phi_m, \phi_m \rangle$, and $\langle \phi_{j_1} \phi_{j_2}, \phi_m \rangle$, the PCE coefficients $y_k$ can be obtained immediately. From the PCE coefficients we can get the moments and compare them to the closed-form expressions.

Notice: A maximum degree of 2 suffices to get the exact solution with PCE. In other words, increasing the maximum degree to values greater than 2 introduces nothing but computational overhead (and numerical errors, possibly).

Practice

First, we create a orthogonal basis relative to $f_X(x)$ of degree at most $d=2$ (degree below).

Notice that we consider a total of Nrec recursion coefficients, and that we also add a quadrature rule by setting addQuadrature = true.

k = 12
using PolyChaos
degree, Nrec = 2, 20
opq = GaussOrthoPoly(degree; Nrec=Nrec, addQuadrature = true);

GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}(2, [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0], GaussMeasure(PolyChaos.w_gaussian, (-Inf, Inf), true), Quad{Float64,Array{Float64,1}}("golubwelsch", 19, [-7.382579024030429, -6.262891156513253, -5.3205363773360395, -4.465872626831033, -3.664416547450638, -2.898051276515753, -2.1555027613169346, -1.4288766760783724, -0.7120850440423797, 9.497049878180818e-17, 0.7120850440423805, 1.4288766760783733, 2.155502761316937, 2.898051276515753, 3.66441654745064, 4.465872626831031, 5.320536377336037, 6.262891156513249, 7.382579024030431], [7.482830054057322e-13, 1.2203708484474712e-9, 2.532220032092887e-7, 1.535114595466668e-5, 0.00037850210941427036, 0.004507235420342024, 0.02866669103011855, 0.10360365727614435, 0.22094171219914369, 0.28377319275152074, 0.220941712199144, 0.10360365727614416, 0.028666691030118426, 0.004507235420342064, 0.0003785021094142682, 1.535114595466658e-5, 2.532220032092883e-7, 1.220370848447473e-9, 7.482830054057216e-13]))

Now let's define a multivariate basis

mop = MultiOrthoPoly([opq for i in 1:k], degree)

MultiOrthoPoly{ProductMeasure,Quad{Float64,Array{Float64,1}},Array{GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}},1}}(["GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}"], 2, 91, [0 0 … 0 0; 1 0 … 0 0; … ; 0 0 … 1 1; 0 0 … 0 2], ProductMeasure(PolyChaos.var"#w#39"{Array{GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}},1}}(GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}[GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}(2, [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0], GaussMeasure(PolyChaos.w_gaussian, (-Inf, Inf), true), Quad{Float64,Array{Float64,1}}("golubwelsch", 19, [-7.382579024030429, -6.262891156513253, -5.3205363773360395, -4.465872626831033, -3.664416547450638, -2.898051276515753, -2.1555027613169346, -1.4288766760783724, -0.7120850440423797, 9.497049878180818e-17, 0.7120850440423805, 1.4288766760783733, 2.155502761316937, 2.898051276515753, 3.66441654745064, 4.465872626831031, 5.320536377336037, 6.262891156513249, 7.382579024030431], [7.482830054057322e-13, 1.2203708484474712e-9, 2.532220032092887e-7, 1.535114595466668e-5, 0.00037850210941427036, 0.004507235420342024, 0.02866669103011855, 0.10360365727614435, 0.22094171219914369, 0.28377319275152074, 0.220941712199144, 0.10360365727614416, 0.028666691030118426, 0.004507235420342064, 0.0003785021094142682, 1.535114595466658e-5, 2.532220032092883e-7, 1.220370848447473e-9, 7.482830054057216e-13])), GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}(2, [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0], GaussMeasure(PolyChaos.w_gaussian, (-Inf, Inf), true), Quad{Float64,Array{Float64,1}}("golubwelsch", 19, [-7.382579024030429, -6.262891156513253, -5.3205363773360395, -4.465872626831033, -3.664416547450638, -2.898051276515753, -2.1555027613169346, -1.4288766760783724, -0.7120850440423797, 9.497049878180818e-17, 0.7120850440423805, 1.4288766760783733, 2.155502761316937, 2.898051276515753, 3.66441654745064, 4.465872626831031, 5.320536377336037, 6.262891156513249, 7.382579024030431], [7.482830054057322e-13, 1.2203708484474712e-9, 2.532220032092887e-7, 1.535114595466668e-5, 0.00037850210941427036, 0.004507235420342024, 0.02866669103011855, 0.10360365727614435, 0.22094171219914369, 0.28377319275152074, 0.220941712199144, 0.10360365727614416, 0.028666691030118426, 0.004507235420342064, 0.0003785021094142682, 1.535114595466658e-5, 2.532220032092883e-7, 1.220370848447473e-9, 7.482830054057216e-13])), GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}(2, [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 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2.532220032092883e-7, 1.220370848447473e-9, 7.482830054057216e-13])), GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}(2, [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0], GaussMeasure(PolyChaos.w_gaussian, (-Inf, Inf), true), Quad{Float64,Array{Float64,1}}("golubwelsch", 19, [-7.382579024030429, -6.262891156513253, -5.3205363773360395, -4.465872626831033, -3.664416547450638, -2.898051276515753, -2.1555027613169346, -1.4288766760783724, -0.7120850440423797, 9.497049878180818e-17, 0.7120850440423805, 1.4288766760783733, 2.155502761316937, 2.898051276515753, 3.66441654745064, 4.465872626831031, 5.320536377336037, 6.262891156513249, 7.382579024030431], [7.482830054057322e-13, 1.2203708484474712e-9, 2.532220032092887e-7, 1.535114595466668e-5, 0.00037850210941427036, 0.004507235420342024, 0.02866669103011855, 0.10360365727614435, 0.22094171219914369, 0.28377319275152074, 0.220941712199144, 0.10360365727614416, 0.028666691030118426, 0.004507235420342064, 0.0003785021094142682, 1.535114595466658e-5, 2.532220032092883e-7, 1.220370848447473e-9, 7.482830054057216e-13])), GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}(2, [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0], GaussMeasure(PolyChaos.w_gaussian, (-Inf, Inf), true), Quad{Float64,Array{Float64,1}}("golubwelsch", 19, [-7.382579024030429, -6.262891156513253, -5.3205363773360395, -4.465872626831033, -3.664416547450638, -2.898051276515753, -2.1555027613169346, -1.4288766760783724, -0.7120850440423797, 9.497049878180818e-17, 0.7120850440423805, 1.4288766760783733, 2.155502761316937, 2.898051276515753, 3.66441654745064, 4.465872626831031, 5.320536377336037, 6.262891156513249, 7.382579024030431], [7.482830054057322e-13, 1.2203708484474712e-9, 2.532220032092887e-7, 1.535114595466668e-5, 0.00037850210941427036, 0.004507235420342024, 0.02866669103011855, 0.10360365727614435, 0.22094171219914369, 0.28377319275152074, 0.220941712199144, 0.10360365727614416, 0.028666691030118426, 0.004507235420342064, 0.0003785021094142682, 1.535114595466658e-5, 2.532220032092883e-7, 1.220370848447473e-9, 7.482830054057216e-13])), GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}(2, [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0], GaussMeasure(PolyChaos.w_gaussian, (-Inf, Inf), true), Quad{Float64,Array{Float64,1}}("golubwelsch", 19, [-7.382579024030429, -6.262891156513253, -5.3205363773360395, -4.465872626831033, -3.664416547450638, 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[7.482830054057322e-13, 1.2203708484474712e-9, 2.532220032092887e-7, 1.535114595466668e-5, 0.00037850210941427036, 0.004507235420342024, 0.02866669103011855, 0.10360365727614435, 0.22094171219914369, 0.28377319275152074, 0.220941712199144, 0.10360365727614416, 0.028666691030118426, 0.004507235420342064, 0.0003785021094142682, 1.535114595466658e-5, 2.532220032092883e-7, 1.220370848447473e-9, 7.482830054057216e-13])), GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}(2, [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0], GaussMeasure(PolyChaos.w_gaussian, (-Inf, Inf), true), Quad{Float64,Array{Float64,1}}("golubwelsch", 19, [-7.382579024030429, -6.262891156513253, -5.3205363773360395, -4.465872626831033, -3.664416547450638, -2.898051276515753, -2.1555027613169346, -1.4288766760783724, -0.7120850440423797, 9.497049878180818e-17, 0.7120850440423805, 1.4288766760783733, 2.155502761316937, 2.898051276515753, 3.66441654745064, 4.465872626831031, 5.320536377336037, 6.262891156513249, 7.382579024030431], [7.482830054057322e-13, 1.2203708484474712e-9, 2.532220032092887e-7, 1.535114595466668e-5, 0.00037850210941427036, 0.004507235420342024, 0.02866669103011855, 0.10360365727614435, 0.22094171219914369, 0.28377319275152074, 0.220941712199144, 0.10360365727614416, 0.028666691030118426, 0.004507235420342064, 0.0003785021094142682, 1.535114595466658e-5, 2.532220032092883e-7, 1.220370848447473e-9, 7.482830054057216e-13])), GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}(2, [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0], GaussMeasure(PolyChaos.w_gaussian, (-Inf, Inf), true), Quad{Float64,Array{Float64,1}}("golubwelsch", 19, [-7.382579024030429, -6.262891156513253, -5.3205363773360395, -4.465872626831033, -3.664416547450638, -2.898051276515753, -2.1555027613169346, -1.4288766760783724, -0.7120850440423797, 9.497049878180818e-17, 0.7120850440423805, 1.4288766760783733, 2.155502761316937, 2.898051276515753, 3.66441654745064, 4.465872626831031, 5.320536377336037, 6.262891156513249, 7.382579024030431], [7.482830054057322e-13, 1.2203708484474712e-9, 2.532220032092887e-7, 1.535114595466668e-5, 0.00037850210941427036, 0.004507235420342024, 0.02866669103011855, 0.10360365727614435, 0.22094171219914369, 0.28377319275152074, 0.220941712199144, 0.10360365727614416, 0.028666691030118426, 0.004507235420342064, 0.0003785021094142682, 1.535114595466658e-5, 2.532220032092883e-7, 1.220370848447473e-9, 7.482830054057216e-13])), GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}(2, [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0], GaussMeasure(PolyChaos.w_gaussian, (-Inf, Inf), true), Quad{Float64,Array{Float64,1}}("golubwelsch", 19, [-7.382579024030429, -6.262891156513253, -5.3205363773360395, -4.465872626831033, -3.664416547450638, -2.898051276515753, -2.1555027613169346, -1.4288766760783724, -0.7120850440423797, 9.497049878180818e-17, 0.7120850440423805, 1.4288766760783733, 2.155502761316937, 2.898051276515753, 3.66441654745064, 4.465872626831031, 5.320536377336037, 6.262891156513249, 7.382579024030431], [7.482830054057322e-13, 1.2203708484474712e-9, 2.532220032092887e-7, 1.535114595466668e-5, 0.00037850210941427036, 0.004507235420342024, 0.02866669103011855, 0.10360365727614435, 0.22094171219914369, 0.28377319275152074, 0.220941712199144, 0.10360365727614416, 0.028666691030118426, 0.004507235420342064, 0.0003785021094142682, 1.535114595466658e-5, 2.532220032092883e-7, 1.220370848447473e-9, 7.482830054057216e-13])), GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}(2, [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0], GaussMeasure(PolyChaos.w_gaussian, (-Inf, Inf), true), Quad{Float64,Array{Float64,1}}("golubwelsch", 19, [-7.382579024030429, -6.262891156513253, -5.3205363773360395, -4.465872626831033, -3.664416547450638, -2.898051276515753, -2.1555027613169346, -1.4288766760783724, -0.7120850440423797, 9.497049878180818e-17, 0.7120850440423805, 1.4288766760783733, 2.155502761316937, 2.898051276515753, 3.66441654745064, 4.465872626831031, 5.320536377336037, 6.262891156513249, 7.382579024030431], [7.482830054057322e-13, 1.2203708484474712e-9, 2.532220032092887e-7, 1.535114595466668e-5, 0.00037850210941427036, 0.004507235420342024, 0.02866669103011855, 0.10360365727614435, 0.22094171219914369, 0.28377319275152074, 0.220941712199144, 0.10360365727614416, 0.028666691030118426, 0.004507235420342064, 0.0003785021094142682, 1.535114595466658e-5, 2.532220032092883e-7, 1.220370848447473e-9, 7.482830054057216e-13]))])

Next, we define the PCE for all $X_i$ with $i = 1, \dots, k$.

L = dim(mop)
mu, sig = 0., 1.
x = [ assign2multi(convert2affinePCE(mu, sig, opq), i, mop.ind) for i in 1:k ]

12-element Array{SparseArrays.SparseVector{Float64,Int64},1}:
   [2 ]  =  1.0
   [3 ]  =  1.0
   [4 ]  =  1.0
   [5 ]  =  1.0
   [6 ]  =  1.0
   [7 ]  =  1.0
   [8 ]  =  1.0
   [9 ]  =  1.0
   [10]  =  1.0
   [11]  =  1.0
   [12]  =  1.0
   [13]  =  1.0

With the orthogonal basis and the quadrature at hand, we can compute the tensors t2 and t3 that store the entries $\langle \phi_m, \phi_m \rangle$, and $\langle \phi_{j_1} \phi_{j_2}, \phi_m \rangle$, respectively.

t2 = Tensor(2,mop)
t3 = Tensor(3,mop)

Tensor{MultiOrthoPoly{ProductMeasure,Quad{Float64,Array{Float64,1}},Array{GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}},1}}}(3,   [1     ]  =  1.0
  [92    ]  =  1.0
  [183   ]  =  1.0
  [274   ]  =  1.0
  [365   ]  =  1.0
  [456   ]  =  1.0
  [547   ]  =  1.0
            ⋮
  [696418]  =  2.0
  [704429]  =  2.0
  [704520]  =  1.0
  [712621]  =  2.0
  [720809]  =  8.0
  [720900]  =  2.0
  [729001]  =  2.0
  [737191]  =  8.0, PolyChaos.var"#getfun#44"{Int64,MultiOrthoPoly{ProductMeasure,Quad{Float64,Array{Float64,1}},Array{GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}},1}},SparseArrays.SparseVector{Float64,Int64}}(3, MultiOrthoPoly{ProductMeasure,Quad{Float64,Array{Float64,1}},Array{GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}},1}}(["GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}", "GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}"], 2, 91, [0 0 … 0 0; 1 0 … 0 0; … ; 0 0 … 1 1; 0 0 … 0 2], ProductMeasure(PolyChaos.var"#w#39"{Array{GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}},1}}(GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}[GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}(2, [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0], GaussMeasure(PolyChaos.w_gaussian, (-Inf, Inf), true), Quad{Float64,Array{Float64,1}}("golubwelsch", 19, [-7.382579024030429, -6.262891156513253, -5.3205363773360395, -4.465872626831033, -3.664416547450638, -2.898051276515753, -2.1555027613169346, -1.4288766760783724, -0.7120850440423797, 9.497049878180818e-17, 0.7120850440423805, 1.4288766760783733, 2.155502761316937, 2.898051276515753, 3.66441654745064, 4.465872626831031, 5.320536377336037, 6.262891156513249, 7.382579024030431], [7.482830054057322e-13, 1.2203708484474712e-9, 2.532220032092887e-7, 1.535114595466668e-5, 0.00037850210941427036, 0.004507235420342024, 0.02866669103011855, 0.10360365727614435, 0.22094171219914369, 0.28377319275152074, 0.220941712199144, 0.10360365727614416, 0.028666691030118426, 0.004507235420342064, 0.0003785021094142682, 1.535114595466658e-5, 2.532220032092883e-7, 1.220370848447473e-9, 7.482830054057216e-13])), GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}(2, [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0], GaussMeasure(PolyChaos.w_gaussian, (-Inf, Inf), true), Quad{Float64,Array{Float64,1}}("golubwelsch", 19, [-7.382579024030429, -6.262891156513253, -5.3205363773360395, -4.465872626831033, -3.664416547450638, -2.898051276515753, -2.1555027613169346, -1.4288766760783724, -0.7120850440423797, 9.497049878180818e-17, 0.7120850440423805, 1.4288766760783733, 2.155502761316937, 2.898051276515753, 3.66441654745064, 4.465872626831031, 5.320536377336037, 6.262891156513249, 7.382579024030431], [7.482830054057322e-13, 1.2203708484474712e-9, 2.532220032092887e-7, 1.535114595466668e-5, 0.00037850210941427036, 0.004507235420342024, 0.02866669103011855, 0.10360365727614435, 0.22094171219914369, 0.28377319275152074, 0.220941712199144, 0.10360365727614416, 0.028666691030118426, 0.004507235420342064, 0.0003785021094142682, 1.535114595466658e-5, 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-0.7120850440423797, 9.497049878180818e-17, 0.7120850440423805, 1.4288766760783733, 2.155502761316937, 2.898051276515753, 3.66441654745064, 4.465872626831031, 5.320536377336037, 6.262891156513249, 7.382579024030431], [7.482830054057322e-13, 1.2203708484474712e-9, 2.532220032092887e-7, 1.535114595466668e-5, 0.00037850210941427036, 0.004507235420342024, 0.02866669103011855, 0.10360365727614435, 0.22094171219914369, 0.28377319275152074, 0.220941712199144, 0.10360365727614416, 0.028666691030118426, 0.004507235420342064, 0.0003785021094142682, 1.535114595466658e-5, 2.532220032092883e-7, 1.220370848447473e-9, 7.482830054057216e-13]))]),   [1     ]  =  1.0
  [92    ]  =  1.0
  [183   ]  =  1.0
  [274   ]  =  1.0
  [365   ]  =  1.0
  [456   ]  =  1.0
  [547   ]  =  1.0
            ⋮
  [696418]  =  2.0
  [704429]  =  2.0
  [704520]  =  1.0
  [712621]  =  2.0
  [720809]  =  8.0
  [720900]  =  2.0
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Quad{Float64,Array{Float64,1}}("golubwelsch", 19, [-7.382579024030429, -6.262891156513253, -5.3205363773360395, -4.465872626831033, -3.664416547450638, -2.898051276515753, -2.1555027613169346, -1.4288766760783724, -0.7120850440423797, 9.497049878180818e-17, 0.7120850440423805, 1.4288766760783733, 2.155502761316937, 2.898051276515753, 3.66441654745064, 4.465872626831031, 5.320536377336037, 6.262891156513249, 7.382579024030431], [7.482830054057322e-13, 1.2203708484474712e-9, 2.532220032092887e-7, 1.535114595466668e-5, 0.00037850210941427036, 0.004507235420342024, 0.02866669103011855, 0.10360365727614435, 0.22094171219914369, 0.28377319275152074, 0.220941712199144, 0.10360365727614416, 0.028666691030118426, 0.004507235420342064, 0.0003785021094142682, 1.535114595466658e-5, 2.532220032092883e-7, 1.220370848447473e-9, 7.482830054057216e-13])), GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}(2, [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0], GaussMeasure(PolyChaos.w_gaussian, (-Inf, Inf), true), Quad{Float64,Array{Float64,1}}("golubwelsch", 19, [-7.382579024030429, -6.262891156513253, -5.3205363773360395, -4.465872626831033, -3.664416547450638, -2.898051276515753, -2.1555027613169346, -1.4288766760783724, -0.7120850440423797, 9.497049878180818e-17, 0.7120850440423805, 1.4288766760783733, 2.155502761316937, 2.898051276515753, 3.66441654745064, 4.465872626831031, 5.320536377336037, 6.262891156513249, 7.382579024030431], [7.482830054057322e-13, 1.2203708484474712e-9, 2.532220032092887e-7, 1.535114595466668e-5, 0.00037850210941427036, 0.004507235420342024, 0.02866669103011855, 0.10360365727614435, 0.22094171219914369, 0.28377319275152074, 0.220941712199144, 0.10360365727614416, 0.028666691030118426, 0.004507235420342064, 0.0003785021094142682, 1.535114595466658e-5, 2.532220032092883e-7, 1.220370848447473e-9, 7.482830054057216e-13])), GaussOrthoPoly{Array{Float64,1},GaussMeasure,Quad{Float64,Array{Float64,1}}}(2, [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0], GaussMeasure(PolyChaos.w_gaussian, (-Inf, Inf), true), Quad{Float64,Array{Float64,1}}("golubwelsch", 19, [-7.382579024030429, -6.262891156513253, -5.3205363773360395, -4.465872626831033, -3.664416547450638, -2.898051276515753, -2.1555027613169346, -1.4288766760783724, -0.7120850440423797, 9.497049878180818e-17, 0.7120850440423805, 1.4288766760783733, 2.155502761316937, 2.898051276515753, 3.66441654745064, 4.465872626831031, 5.320536377336037, 6.262891156513249, 7.382579024030431], [7.482830054057322e-13, 1.2203708484474712e-9, 2.532220032092887e-7, 1.535114595466668e-5, 0.00037850210941427036, 0.004507235420342024, 0.02866669103011855, 0.10360365727614435, 0.22094171219914369, 0.28377319275152074, 0.220941712199144, 0.10360365727614416, 0.028666691030118426, 0.004507235420342064, 0.0003785021094142682, 1.535114595466658e-5, 2.532220032092883e-7, 1.220370848447473e-9, 7.482830054057216e-13]))]))

With the tensors at hand, we can compute the Galerkin projection.

Notice: there are more efficient ways to do this, but let's keep it simple.

y = [ sum( x[i][j1]*x[i][j2]*t3.get([j1-1,j2-1,m-1])/t2.get([m-1,m-1])  for i=1:k, j1=1:L, j2=1:L ) for m=1:L ]

91-element Array{Float64,1}:
 12.0
  0.0
  0.0
  0.0
  0.0
  0.0
  0.0
  0.0
  0.0
  0.0
  ⋮
  0.0
  0.0
  0.0
  1.0000000000000016
  0.0
  0.0
  1.0000000000000016
  0.0
  1.0000000000000016

Let's compare the moments via PCE to the closed-form expressions.

moms_analytic(k) = [k, sqrt(2k), sqrt(8/k)]
function myskew(y)
   e3 = sum( y[i]*y[j]*y[k]*t3.get([i-1,j-1,k-1]) for i=1:L,j=1:L,k=1:L )
   μ = y[1]
   σ = std(y,mop)
   (e3-3*μ*σ^2-μ^3)/(σ^3)
end

print("Expected value:\t\t$(moms_analytic(k)[1]) = $(mean(y,mop))\n")
print("\t\t\terror = $(abs(mean(y,mop)-moms_analytic(k)[1]))\n")
print("Standard deviation:\t$(moms_analytic(k)[2]) = $(std(y,mop))\n")
print("\t\t\terror = $(moms_analytic(k)[2]-std(y,mop))\n")
print("Skewness:\t\t$(moms_analytic(k)[3]) = $(myskew(y))\n")
print("\t\t\terror = $(moms_analytic(k)[3]-myskew(y))\n")

Expected value:		12.0 = 12.0
			error = 0.0
Standard deviation:	4.898979485566356 = 4.898979485566365
			error = -8.881784197001252e-15
Skewness:		0.816496580927726 = 0.8164965809276967
			error = 2.930988785010413e-14

Let's plot the probability density function to compare results. We first draw samples from the measure with the help of sampleMeasure(), and then evaluate the basis at these samples and multiply times the PCE coefficients. The latter stop is done using evaluatePCE(). Both steps are combined in the function samplePCE(). Finally, we compare the result agains the analytical PDF $\rho(t) = \frac{t^{t/2-1}\mathrm{e}^{-t/2}}{2^{k/2} \, \Gamma(k/2)}$ of the chi-squared distribution with one degree of freedom.

using Plots
Nsmpl = 10000
# long way: ξ = sampleMeasure(Nsmpl,mop), ysmpl = evaluatePCE(y,ξ,mop)
ysmpl = samplePCE(Nsmpl, y, mop)
histogram(ysmpl;normalize=true, xlabel="t",ylabel="rho(t)")

import SpecialFunctions: gamma
ρ(t) = 1/(2^(0.5*k)*gamma(0.5*k))*t^(0.5*k-1)*exp(-0.5*t)
t = range(0.1; stop=maximum(ysmpl), length=100)
plot!(t, ρ.(t), w=4)