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use anyhow::Result;
use num::Float;
use std::cmp::min;
use std::convert::From;
use std::fmt::Debug;
use std::ops::{Add, AddAssign, Div};
use finitediff::FiniteDiff;
use liblbfgs::lbfgs;
use crate::{acf, util};
/// Calculate residuals given a time series, an intercept, and ARMA parameters
/// phi and theta. Any differencing and centering should be done before.
/// Squaring and summing the residuals yields the conditional sum of squares (CSS),
/// which can be used as an objective function to estimate the AR and MA parameters.
/// The variance can be then estimated via `CSS/(x.len()-phi.len())`.
///
/// # Arguments
///
/// * `&x` - Vector of the timeseries.
/// * `intercept` - Intercept parameter.
/// * `&phi` - AR parameter vector.
/// * `&theta` - MA parameter vector.
///
/// # Returns
///
/// * Vector of residuals. The first `phi.len()` items are zeros.
///
/// # Example
///
/// ```
/// use arima::estimate;
/// let x = [1.0, 1.2, 1.4, 1.6];
/// let res = estimate::residuals(&x, 0.0, Some(&[0.6, 0.4]), Some(&[0.3])).unwrap();
/// assert!((res[0] - 0.00).abs() < 1.0e-7);
/// assert!((res[1] - 0.00).abs() < 1.0e-7);
/// assert!((res[2] - 0.27999999).abs() < 1.0e-7);
/// assert!((res[3] - 0.196).abs() < 1.0e-7);
/// ```
pub fn residuals<T: Float + From<u32> + From<f64> + Copy + Add + AddAssign + Div + Debug>(
x: &[T],
intercept: T,
phi: Option<&[T]>,
theta: Option<&[T]>,
) -> Result<Vec<T>> {
let phi = phi.unwrap_or(&[]);
let theta = theta.unwrap_or(&[]);
if x.len() < phi.len() || x.len() < theta.len() {
anyhow::bail!("Too many items in phi or theta");
}
let zero: T = From::from(0.0);
let mut residuals: Vec<T> = Vec::new();
for _ in 0..phi.len() {
residuals.push(zero);
}
for t in phi.len()..x.len() {
let mut xt: T = intercept;
for j in 0..phi.len() {
xt += phi[j] * x[t - j - 1];
}
for j in 0..min(theta.len(), t) {
xt += theta[j] * residuals[t - j - 1];
}
residuals.push(x[t] - xt);
}
Ok(residuals)
}
/// Fit an ARIMA model. Returns the fitted coefficients.
/// This method uses the L-BFGS algorithm and the conditional sum of squares (CSS)
/// as the objective function.
///
/// # Arguments
///
/// * `&x` - Vector of the timeseries.
/// * `ar` - Order of the AR coefficients.
/// * `d` - Order of differencing.
/// * `ma` - Order of the MA coefficients.
///
/// # Returns
///
/// * ARIMA coefficients minimizing the conditional sum of squares (CSS).
///
/// # Example
///
/// ```
/// use arima::estimate;
/// let x = [1.0, 1.2, 1.4, 1.6, 1.4, 1.2, 1.0];
/// let coef = estimate::fit(&x, 0, 0, 1).unwrap();
/// assert!((coef[0] - 1.2051).abs() < 1.0e-3); // intercept
/// assert!((coef[1] - 0.5637).abs() < 1.0e-3); // phi_1
/// ```
pub fn fit<T: Float + From<u32> + From<f64> + Into<f64> + Copy + Add + AddAssign + Div + Debug>(
x: &[T],
ar: usize,
d: usize,
ma: usize,
) -> Result<Vec<f64>> {
// Convert into f64 as the optimizer functions only support f64
let mut x64: Vec<f64> = Vec::new();
for a in x {
x64.push((*a).into());
}
let mut x = x64;
if d > 0 {
x = util::diff(&x, d);
}
let x = x;
let total_size = 1 + ar + ma;
// The objective is to minimize the conditional sum of squares (CSS),
// i.e. the sum of the squared residuals
let f = |coef: &Vec<f64>| {
assert_eq!(coef.len(), total_size);
let intercept = coef[0];
let phi = &coef[1..ar + 1];
let theta = &coef[ar + 1..];
let residuals = residuals(&x, intercept, Some(phi), Some(theta)).unwrap();
let mut css: f64 = 0.0;
for residual in &residuals {
css += residual * residual;
}
css
};
let g = |coef: &Vec<f64>| coef.forward_diff(&f);
// Initial coefficients
// Todo: These initial guesses are rather arbitrary.
let mut coef: Vec<f64> = Vec::new();
// Initial guess for the intercept: First value of x
coef.push(util::mean(&x));
// Initial guess for the AR coefficients: Values of the PACF
if ar > 0 {
let pacf = acf::pacf(&x, Some(ar)).unwrap();
for p in pacf {
coef.push(p);
}
}
// Initial guess for the MA coefficients: 1.0
if ma > 0 {
coef.resize(coef.len() + ma, 1.0);
}
let evaluate = |x: &[f64], gx: &mut [f64]| {
let x_vec = x.to_vec();
let fx = f(&x_vec);
let gx_eval = g(&x_vec);
// copy values from gx_eval into gx
gx[..gx_eval.len()].copy_from_slice(&gx_eval[..]);
Ok(fx)
};
let fmin = lbfgs().with_max_iterations(200);
if let Err(e) = fmin.minimize(
&mut coef, // input variables
evaluate, // define how to evaluate function
|_prgr| {
false // returning true will cancel optimization
},
) {
tracing::warn!("Got error during fit: {}", e);
}
Ok(coef)
}
/// TODO clean up
/// Auto-fit an ARIMA model, guessing AR and MA orders.
/// See `fit` for more details.
///
/// # Arguments
///
/// * `&x` - Vector of the timeseries.
/// * `d` - Order of differencing.
///
/// # Returns
///
/// * ARIMA coefficients minimizing the conditional sum of squares (CSS).
pub fn autofit<
T: Float + From<u32> + From<f64> + Into<f64> + Copy + Add + AddAssign + Div + Debug,
>(
x: &[T],
d: usize,
) -> Result<Vec<f64>> {
let x: Vec<f64> = x.iter().map(|v| (*v).into()).collect();
let n = x.len() as f64;
let n_lags = 12;
// Hardcoding for now
// let alpha = 0.05;
// ppf = scipy.stats.norm.ppf(1 - alpha / 2.0)
let ppf = 1.959963984540054;
// Estimate MA order
// <https://www.statsmodels.org/devel/_modules/statsmodels/tsa/stattools.html#acf>
let _acf = acf::acf(&x, Some(n_lags), false).unwrap();
let mult: Vec<f64> = _acf[1.._acf.len() - 1]
.iter()
.scan(0., |acc, v| {
*acc += v.powf(2.);
Some(1. + 2. * *acc)
})
.collect();
let mut varacf = vec![0., 1. / n];
let varacf_end: Vec<f64> = (0.._acf.len() - 2).map(|i| 1. / n * mult[i]).collect();
varacf.extend(varacf_end);
let interval: Vec<f64> = varacf.iter().map(|v| ppf * v.sqrt()).collect();
let confint: Vec<(f64, f64)> = _acf
.iter()
.zip(&interval)
.map(|(p, q)| (p - q, p + q))
.collect();
let bounds: Vec<(f64, f64)> = confint
.iter()
.zip(&_acf)
.map(|((l, u), a)| (l - a, u - a))
.collect();
// Subtract one to compensate for the first value (lag=0)
let ma_order = _acf
.iter()
.zip(bounds)
.take_while(|(a, (l, u))| a < &l || a > &u)
.count()
- 1;
// <https://www.statsmodels.org/devel/_modules/statsmodels/tsa/stattools.html#pacf>
let _pacf = acf::pacf(&x, Some(n_lags)).unwrap();
let pacf_varacf = 1.0 / n;
let pacf_interval = ppf * pacf_varacf.sqrt();
let pacf_confint: Vec<(f64, f64)> = _pacf
.iter()
.map(|p| (p - pacf_interval, p + pacf_interval))
.collect();
let pacf_bounds: Vec<(f64, f64)> = pacf_confint
.iter()
.zip(&_pacf)
.map(|((l, u), a)| (l - a, u - a))
.collect();
// lag=0 isn't included so no need to subtract one
let ar_order = _pacf
.iter()
.zip(pacf_bounds)
.take_while(|(a, (l, u))| a < &l || a > &u)
.count();
fit(&x, ar_order, d, ma_order)
}