Sensitivity Analysis - Research Notebook

Gradient-Based Sensitivity Analysis¶

Spatial Field¶

Consider an arbitrary spatial field

\begin{aligned} u &= \boldsymbol{u}(\mathbf{s}) && && u: \mathbb{R}^{D_s}\rightarrow\mathbb{R} && && \mathbf{s}\in\Omega\subseteq\mathbb{R}^{D_s} \end{aligned}

(1)

We can take the gradient of that field

\begin{aligned} \text{Gradient}: && && \boldsymbol{J}[\boldsymbol{u}, \mathbf{s}] &= \partial_s \boldsymbol{u} && && \partial_s \boldsymbol{u}: \mathbb{R}^{D_s}\rightarrow\mathbb{R}^{D_s} \end{aligned}

(2)

Example¶

An example would be a simple sine with an additive term

\begin{aligned} u(s) &= s + \sin(s) && && u: \mathbb{R}\rightarrow \mathbb{R} \end{aligned}

(3)

# get data
x: Array["N"] = ...
# create a function
f = lambda x: x + sin(x)
y: Array["N"] = vmap(f)(x)

Now, we can take the gradient of that field.

\begin{aligned} \partial_s u &= \cos(s) && && u: \mathbb{R}\rightarrow \mathbb{R} \end{aligned}

(4)

# take the gradient 
df = grad(f)
dx: Array["N"] = vmap(df)(x)

Parameterized Spatial Field¶

Suppose we have some sample from the arbitrary spatial field with some parameters

\begin{aligned} u &= \boldsymbol{u}(\mathbf{s},\boldsymbol{\theta}) && && u: \mathbb{R}^{D_s}\times\mathbb{R}^{D_\theta}\rightarrow\mathbb{R} && && \mathbf{s}\in\Omega\subseteq\mathbb{R}^{D_s} \end{aligned}

(5)

Now, we have a few options. We can take the gradient of

\begin{aligned} \text{Gradient wrt Coords}: && && \boldsymbol{J}[\boldsymbol{u}, \mathbf{s}] &= \partial_s \boldsymbol{u} && && \partial_s \boldsymbol{u}: \mathbb{R}^{D_s}\times\mathbb{R}^{D_\theta}\rightarrow\mathbb{R}^{D_s}\\ \text{Gradient wrt Parameters}: && && \boldsymbol{J}[\boldsymbol{u}, \mathbf{\theta}] &= \partial_\theta \boldsymbol{u} && && \partial_\theta \boldsymbol{u}: \mathbb{R}^{D_s}\times\mathbb{R}^{D_\theta}\rightarrow\mathbb{R}^{D_\theta}\\ \end{aligned}

(6)

Example¶

An example would be a simple periodic sine function with some amplitude and frequency parameters

\begin{aligned} u(s;\boldsymbol{\theta}) &= a\sin(bs + c) + d && && u: \mathbb{R}\times\mathbb{R}^{D_\theta}\rightarrow \mathbb{R} \end{aligned}

(7)

where $a$ is the amplitude, $b$ is the period, $c$ is the phase shift and $d$ is the vertical shift. So the parameters of the field are $\boldsymbol{\theta}= \{a,b,c,d\}$ . See webpage for a visual demonstration.

# create a function
def f(x: Scalar, params: Dict) -> Scalar:
    a, b, c, d = params["a"], params["b"], params["c"], params["d"]
    return a sin(b * s + c) + d

# Define parameters
params = dict(a=0.1,b=0.01,c=1.0,d=10)
# get data
x: Array["N"] = np.linspace(-2*pi, 2*pi, 100)

# apply function
y: Array["N"] = vmap(f, axis=0)(x, params)

Now, we can take the gradient of that field. First, the gradient wrt the coordinate, $s$ .

\begin{aligned} \text{Gradient wrt Coords}: && && \partial_s \boldsymbol{u} &= ab\cos(bs+d) && && \partial_s \boldsymbol{u}: \mathbb{R}\times\mathbb{R}^{D_\theta}\rightarrow\mathbb{R} \end{aligned}

(8)

Now, we can take the gradient wrt each of the parameters

\begin{aligned} \text{Gradient wrt }a: && && \partial_a \boldsymbol{u} &= \sin(bs + c) && && \partial_a \boldsymbol{u}: \mathbb{R}\times\mathbb{R}^{D_\theta}\rightarrow\mathbb{R} \\ \text{Gradient wrt }b: && && \partial_b \boldsymbol{u} &= as\cos(bs+c) && && \partial_b \boldsymbol{u}: \mathbb{R}\times\mathbb{R}^{D_\theta}\rightarrow\mathbb{R} \\ \text{Gradient wrt }c: && && \partial_c \boldsymbol{u} &= a\cos(bs+c) && && \partial_c \boldsymbol{u}: \mathbb{R}\times\mathbb{R}^{D_\theta}\rightarrow\mathbb{R} \\ \text{Gradient wrt }d: && && \partial_d \boldsymbol{u} &= 1 && && \partial_d \boldsymbol{u}: \mathbb{R}\times\mathbb{R}^{D_\theta}\rightarrow\mathbb{R} \\ \end{aligned}

(9)

# take the gradient 
dxf = grad(f, axis=0)
dx: Array["N"] = vmap(dxf, axis=0)(x)

# gradient wrt parameters
dpf = jacobian(f, axis=1)
dp: Array["N Ds"] = vmap(dpf, axis=1)(x)