Splines

Spline models and samples

spline_component

 spline_component (knots:Union[int,List[~T]], index:Iterable[~T],
                   degree:int=3)

	Type	Default	Details
knots	Union		Number of knots or interior knots to use
index	Iterable		index
degree	int	3	Knot degree defaults to cubic splines

Example 1 (If knots are an integer)

N_SPLINES = 9
INDEX = [i for i in range(156)]
splines = spline_component(
    N_SPLINES,
    INDEX
)

Figure 2: Samples of linear combination of splines.

Example 2 (Knots can be directly specified)

INDEX = pd.date_range('01/01/2021', periods=156, freq="W-MON")
KNOTS = [INDEX[10], INDEX[30], INDEX[60], INDEX[120]]
splines_date = spline_component(
    list(map(INDEX.get_loc, KNOTS)),
    np.arange(len(INDEX))
)

Example 3 (Spline Regression Model)

INDEX = np.linspace(0, 1, 156)

spline_data = spline_component(19, INDEX, degree=3) # 19 Splines
fun_ = lambda x: np.sin(2*np.pi*5*x)/(20*(x-.5)) + (x-.5) * 2
fun = fun_(INDEX)# Complex Non-linear function to learn

y_obs = fun + np.random.normal(0, .1, size=spline_data.shape[0]) # Noisy observation process

tau_scale = 1.0
c_scale = 0.1
with pm.Model() as model:
    tau = pm.HalfCauchy('tau', tau_scale)
    lambdas = pm.HalfCauchy("lambdas", 1, shape=spline_data.shape[1])
    c = pm.HalfCauchy("c", beta=c_scale)

    sigma_coeff = pm.Deterministic("sigma_coeff", tau * lambdas / pm.math.sqrt(c**2 + (tau * lambdas)**2))
    betas_ = pm.Normal("betas_", 0, sigma_coeff)
    betas = pm.Deterministic("betas", pm.math.cumsum(betas_)) # Enforce Random Walk Process

    alpha = pm.Normal("alpha", 0, 1)

    spline_trend = pm.Deterministic("splines", spline_data@betas + alpha)
    precision = pm.HalfCauchy("precision", 2)
    pm.Normal("mu", mu=spline_trend, tau=precision, observed=y_obs)

Figure 3: Graph of prior samples from the spline regression model.

Table 1: Summary of coefficients for the spline regression model.

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
betas[0]	0.009	0.131	-0.243	0.262	0.003	0.002	1407.0	2094.0	1.0
betas[1]	-0.063	0.118	-0.281	0.159	0.003	0.002	2027.0	2518.0	1.0
betas[2]	0.378	0.114	0.173	0.598	0.003	0.002	1621.0	2392.0	1.0
betas[3]	0.548	0.112	0.324	0.743	0.003	0.002	1845.0	2276.0	1.0
betas[4]	0.389	0.109	0.177	0.590	0.003	0.002	1625.0	2538.0	1.0
betas[5]	0.197	0.113	-0.034	0.391	0.003	0.002	1607.0	2097.0	1.0
betas[6]	0.977	0.111	0.757	1.175	0.003	0.002	1613.0	1949.0	1.0
betas[7]	1.206	0.113	1.009	1.435	0.003	0.002	1537.0	2096.0	1.0
betas[8]	0.692	0.109	0.469	0.883	0.002	0.002	1924.0	2232.0	1.0
betas[9]	-0.551	0.107	-0.756	-0.350	0.003	0.002	1737.0	2105.0	1.0
betas[10]	-0.544	0.106	-0.734	-0.341	0.002	0.002	2018.0	2124.0	1.0
betas[11]	1.054	0.110	0.851	1.265	0.002	0.002	1973.0	2353.0	1.0
betas[12]	1.833	0.111	1.622	2.037	0.003	0.002	1930.0	2226.0	1.0
betas[13]	1.488	0.111	1.281	1.702	0.003	0.002	1957.0	2474.0	1.0
betas[14]	1.369	0.104	1.173	1.565	0.003	0.002	1728.0	2645.0	1.0
betas[15]	1.458	0.109	1.247	1.655	0.002	0.002	2259.0	2227.0	1.0
betas[16]	2.077	0.109	1.865	2.279	0.003	0.002	1800.0	1994.0	1.0
betas[17]	1.644	0.133	1.367	1.864	0.003	0.002	2406.0	2565.0	1.0
betas[18]	1.939	0.129	1.687	2.169	0.003	0.002	2225.0	2990.0	1.0
betas[19]	2.060	0.106	1.875	2.274	0.002	0.001	2802.0	3035.0	1.0
alpha	-1.054	0.078	-1.200	-0.905	0.002	0.001	2102.0	2160.0	1.0
tau	1.020	5.563	0.003	2.581	0.113	0.080	2561.0	2382.0	1.0

Figure 4: Posterior for horseshoe regularized spline regression.