README.md

中文博客主页：https://blog.csdn.net/linjing_zyq

pip install optimtool

1. 无约束优化算法性能对比

前五个参数完全一致，其中第四个参数是绘图接口，默认绘制单个算法的迭代过程；第五个参数是输出函数迭代值接口，默认为不输出。

method：用于传递线搜索方式

from optimtool.unconstrain import gradient_descent

方法	函数参数	调用示例
解方程得到精确解法（solve）	`solve(funcs, args, x_0, draw=True, output_f=False, epsilon=1e-10, k=0)`	gradient_descent.solve(funcs, args, x_0)
基于Grippo非单调线搜索的梯度下降法	`barzilar_borwein(funcs, args, x_0, draw=True, output_f=False, method="grippo", M=20, c1=0.6, beta=0.6, alpha=1, epsilon=1e-10, k=0)`	gradient_descent.barzilar_borwein(funcs, args, x_0, method="grippo")
基于ZhangHanger非单调线搜索的梯度下降法	`barzilar_borwein(funcs, args, x_0, draw=True, output_f=False, method="ZhangHanger", M=20, c1=0.6, beta=0.6, alpha=1, epsilon=1e-10, k=0)`	gradient_descent.barzilar_borwein(funcs, args, x_0, method="ZhangHanger")
基于最速下降法的梯度下降法	`steepest(funcs, args, x_0, draw=True, output_f=False, method="wolfe", epsilon=1e-10, k=0)`	gradient_descent.steepest(funcs, args, x_0)

from optimtool.unconstrain import newton

方法	函数参数	调用示例
经典牛顿法	`classic(funcs, args, x_0, draw=True, output_f=False, epsilon=1e-10, k=0)`	newton.classic(funcs, args, x_0)
基于armijo线搜索方法的修正牛顿法	`modified(funcs, args, x_0, draw=True, output_f=False, method="armijo", m=20, epsilon=1e-10, k=0)`	newton.modified(funcs, args, x_0, method="armijo")
基于goldstein线搜索方法的修正牛顿法	`modified(funcs, args, x_0, draw=True, output_f=False, method="goldstein", m=20, epsilon=1e-10, k=0)`	newton.modified(funcs, args, x_0, method="goldstein")
基于wolfe线搜索方法的修正牛顿法	`modified(funcs, args, x_0, draw=True, output_f=False, method="wolfe", m=20, epsilon=1e-10, k=0)`	newton.modified(funcs, args, x_0, method="wolfe")
基于armijo线搜索方法的非精确牛顿法	`CG(funcs, args, x_0, draw=True, output_f=False, method="armijo", epsilon=1e-6, k=0)`	newton.CG(funcs, args, x_0, method="armijo")
基于goldstein线搜索方法的非精确牛顿法	`CG(funcs, args, x_0, draw=True, output_f=False, method="goldstein", epsilon=1e-6, k=0)`	newton.CG(funcs, args, x_0, method="goldstein")
基于wolfe线搜索方法的非精确牛顿法	`CG(funcs, args, x_0, draw=True, output_f=False, method="wolfe", epsilon=1e-6, k=0)`	newton.CG(funcs, args, x_0, method="wolfe")

from optimtool.unconstrain import newton_quasi

方法	函数参数	调用示例
基于BFGS方法更新海瑟矩阵的拟牛顿法	`bfgs(funcs, args, x_0, draw=True, output_f=False, method="wolfe", m=20, epsilon=1e-10, k=0)`	newton_quasi.bfgs(funcs, args, x_0)
基于DFP方法更新海瑟矩阵的拟牛顿法	`dfp(funcs, args, x_0, draw=True, output_f=False, method="wolfe", m=20, epsilon=1e-4, k=0)`	newton_quasi.dfp(funcs, args, x_0)
基于有限内存BFGS方法更新海瑟矩阵的拟牛顿法	`L_BFGS(funcs, args, x_0, draw=True, output_f=False, method="wolfe", m=6, epsilon=1e-10, k=0)`	newton_quasi.L_BFGS(funcs, args, x_0)

from optimtool.unconstrain import trust_region

方法	函数参数	调用示例
基于截断共轭梯度法的信赖域算法	`steihaug_CG(funcs, args, x_0, draw=True, output_f=False, m=100, r0=1, rmax=2, eta=0.2, p1=0.4, p2=0.6, gamma1=0.5, gamma2=1.5, epsilon=1e-6, k=0)`	trust_region.steihaug_CG(funcs, args, x_0)

import sympy as sp
import matplotlib.pyplot as plt
import optimtool as oo

f, x1, x2, x3, x4 = sp.symbols("f x1 x2 x3 x4")
f = (x1 - 1)**2 + (x2 - 1)**2 + (x3 - 1)**2 + (x1**2 + x2**2 + x3**2 + x4**2 - 0.25)**2
funcs = sp.Matrix([f])
args = sp.Matrix([x1, x2, x3, x4])
x_0 = (1, 2, 3, 4)

# 无约束优化测试函数性能对比
f_list = []
title = ["gradient_descent_barzilar_borwein", "newton_CG", "newton_quasi_L_BFGS", "trust_region_steihaug_CG"]
colorlist = ["maroon", "teal", "slateblue", "orange"]
_, _, f = oo.unconstrain.gradient_descent.barzilar_borwein(funcs, args, x_0, False, True)
f_list.append(f)
_, _, f = oo.unconstrain.newton.CG(funcs, args, x_0, False, True)
f_list.append(f)
_, _, f = oo.unconstrain.newton_quasi.L_BFGS(funcs, args, x_0, False, True)
f_list.append(f)
_, _, f = oo.unconstrain.trust_region.steihaug_CG(funcs, args, x_0, False, True)
f_list.append(f)

# 绘图
handle = []
for j, z in zip(colorlist, f_list):
    ln, = plt.plot([i for i in range(len(z))], z, c=j, marker='o', linestyle='dashed')
    handle.append(ln)
plt.xlabel("$Iteration \ times \ (k)$")
plt.ylabel("$Objective \ function \ value: \ f(x_k)$")
plt.legend(handle, title)
plt.title("Performance Comparison")
plt.show()

2. 非线性最小二乘问题

from optimtool.unconstrain import nonlinear_least_square

method：用于传递线搜索方法

方法	函数参数	调用示例
基于高斯牛顿法的非线性最小二乘问题解法	`gauss_newton(funcr, args, x_0, draw=True, output_f=False, method="wolfe", epsilon=1e-10, k=0)`	nonlinear_least_square.gauss_newton(funcr, args, x_0)
基于levenberg_marquardt的非线性最小二乘问题解法	`levenberg_marquardt(funcr, args, x_0, draw=True, output_f=False, m=100, lamk=1, eta=0.2, p1=0.4, p2=0.9, gamma1=0.7, gamma2=1.3, epsilon=1e-10, k=0)`	nonlinear_least_square.levenberg_marquardt(funcr, args, x_0)

import sympy as sp
import matplotlib.pyplot as plt
import optimtool as oo

r1, r2, x1, x2 = sp.symbols("r1 r2 x1 x2")
r1 = x1**3 - 2*x2**2 - 1
r2 = 2*x1 + x2 - 2
funcr = sp.Matrix([r1, r2])
args = sp.Matrix([x1, x2])
x_0 = (2, 2)

f_list = []
title = ["gauss_newton", "levenberg_marquardt"]
colorlist = ["maroon", "teal"]
_, _, f = oo.unconstrain.nonlinear_least_square.gauss_newton(funcr, args, x_0, False, True) # 第五参数控制输出函数迭代值列表
f_list.append(f)
_, _, f = oo.unconstrain.nonlinear_least_square.levenberg_marquardt(funcr, args, x_0, False, True)
f_list.append(f)

# 绘图
handle = []
for j, z in zip(colorlist, f_list):
    ln, = plt.plot([i for i in range(len(z))], z, c=j, marker='o', linestyle='dashed')
    handle.append(ln)
plt.xlabel("$Iteration \ times \ (k)$")
plt.ylabel("$Objective \ function \ value: \ f(x_k)$")
plt.legend(handle, title)
plt.title("Performance Comparison")
plt.show()

3. 等式约束优化测试

from optimtool.constrain import equal

无约束内核默认采用wolfe线搜索方法

方法	函数参数	调用示例
二次罚函数法	`penalty_quadratic(funcs, args, cons, x_0, draw=True, output_f=False, method="gradient_descent", sigma=10, p=2, epsilon=1e-4, k=0)`	equal.penalty_quadratic(funcs, args, cons, x_0)
增广拉格朗日法	`lagrange_augmented(funcs, args, cons, x_0, draw=True, output_f=False, method="gradient_descent", lamk=6, sigma=10, p=2, etak=1e-4, epsilon=1e-6, k=0)`	equal.lagrange_augmented(funcs, args, cons, x_0)

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt
import optimtool as oo

f, x1, x2 = sp.symbols("f x1 x2")
f = x1 + np.sqrt(3) * x2
c1 = x1**2 + x2**2 - 1
funcs = sp.Matrix([f])
cons = sp.Matrix([c1])
args = sp.Matrix([x1, x2])
x_0 = (-1, -1)

f_list = []
title = ["penalty_quadratic", "lagrange_augmented"]
colorlist = ["maroon", "teal"]
_, _, f = oo.constrain.equal.penalty_quadratic(funcs, args, cons, x_0, False, True) # 第四个参数控制单个算法不显示迭代图，第五参数控制输出函数迭代值列表
f_list.append(f)
_, _, f = oo.constrain.equal.lagrange_augmented(funcs, args, cons, x_0, False, True)
f_list.append(f)

# 绘图
handle = []
for j, z in zip(colorlist, f_list):
    ln, = plt.plot([i for i in range(len(z))], z, c=j, marker='o', linestyle='dashed')
    handle.append(ln)
plt.xlabel("$Iteration \ times \ (k)$")
plt.ylabel("$Objective \ function \ value: \ f(x_k)$")
plt.legend(handle, title)
plt.title("Performance Comparison")
plt.show()

4. 不等式约束优化测试

from optimtool.constrain import unequal

无约束内核默认采用wolfe线搜索方法

方法	函数参数	调用示例
二次罚函数法	`penalty_quadratic(funcs, args, cons, x_0, draw=True, output_f=False, method="gradient_descent", sigma=1, p=0.4, epsilon=1e-10, k=0)`	unequal.penalty_quadratic(funcs, args, cons, x_0)
内点（分式）罚函数法	`penalty_interior_fraction(funcs, args, cons, x_0, draw=True, output_f=False, method="gradient_descent", sigma=12, p=0.6, epsilon=1e-6, k=0)`	unequal.penalty_interior_fraction(funcs, args, cons, x_0)
拉格朗日法（本质上为不存在等式约束）	`lagrange_augmented(funcs, args, cons, x_0, draw=True, output_f=False, method="gradient_descent", muk=10, sigma=8, alpha=0.2, beta=0.7, p=2, eta=1e-1, epsilon=1e-4, k=0)`	unequal.lagrange_augmented(funcs, args, cons, x_0)

import sympy as sp
import matplotlib.pyplot as plt
import optimtool as oo

f, x1, x2 = sp.symbols("f x1 x2")
f = x1**2 + (x2 - 2)**2
c1 = 1 - x1
c2 = 2 - x2
funcs = sp.Matrix([f])
cons = sp.Matrix([c1, c2])
args = sp.Matrix([x1, x2])
x_0 = (2, 3)

f_list = []
title = ["penalty_quadratic", "penalty_interior_fraction"]
colorlist = ["maroon", "teal"]
_, _, f = oo.constrain.unequal.penalty_quadratic(funcs, args, cons, x_0, False, True, method="newton", sigma=10, epsilon=1e-6) # 第四个参数控制单个算法不显示迭代图，第五参数控制输出函数迭代值列表
f_list.append(f)
_, _, f = oo.constrain.unequal.penalty_interior_fraction(funcs, args, cons, x_0, False, True, method="newton")
f_list.append(f)

# 绘图
handle = []
for j, z in zip(colorlist, f_list):
    ln, = plt.plot([i for i in range(len(z))], z, c=j, marker='o', linestyle='dashed')
    handle.append(ln)
plt.xlabel("$Iteration \ times \ (k)$")
plt.ylabel("$Objective \ function \ value: \ f(x_k)$")
plt.legend(handle, title)
plt.title("Performance Comparison")
plt.show()

单独测试拉格朗日方法：

# 导入符号运算的包
import sympy as sp

# 导入约束优化
import optimtool as oo

# 构造函数
f1 = sp.symbols("f1")
x1, x2, x3, x4 = sp.symbols("x1 x2 x3 x4")
f1 = x1**2 + x2**2 + 2*x3**3 + x4**2 - 5*x1 - 5*x2 - 21*x3 + 7*x4
c1 = 8 - x1 + x2 - x3 + x4 - x1**2 - x2**2 - x3**2 - x4**2
c2 = 10 + x1 + x4 - x1**2 - 2*x2**2 - x3**2 - 2*x4**2
c3 = 5 - 2*x1 + x2 + x4 - 2*x1**2 - x2**2 - x3**2
cons_unequal1 = sp.Matrix([c1, c2, c3])
funcs1 = sp.Matrix([f1])
args1 = sp.Matrix([x1, x2, x3, x4])
x_1 = (0, 0, 0, 0)

x_0, _, f = oo.constrain.unequal.lagrange_augmented(funcs1, args1, cons_unequal1, x_1, output_f=True, method="trust_region", sigma=1, muk=1, p=1.2)
for i in range(len(x_0)):
     x_0[i] = round(x_0[i], 2)
print("\n最终收敛点：", x_0, "\n目标函数值：", f[-1])

result：

最终收敛点： [ 2.5   2.5   1.87 -3.5 ] 
目标函数值： -50.94151192711454

5. 混合等式约束测试

from optimtool.constrain import mixequal

无约束内核默认采用wolfe线搜索方法

方法	函数参数	调用示例
二次罚函数法	`penalty_quadratic(funcs, args, cons_equal, cons_unequal, x_0, draw=True, output_f=False, method="gradient_descent", sigma=1, p=0.6, epsilon=1e-10, k=0)`	mixequal.penalty_quadratic(funcs, args, cons_equal, cons_unequal, x_0)
L1罚函数法	`penalty_L1(funcs, args, cons_equal, cons_unequal, x_0, draw=True, output_f=False, method="gradient_descent", sigma=1, p=0.6, epsilon=1e-10, k=0)`	mixequal.penalty_L1(funcs, args, cons_equal, cons_unequal, x_0)
增广拉格朗日函数法	`lagrange_augmented(funcs, args, cons_equal, cons_unequal, x_0, draw=True, output_f=False, method="gradient_descent", lamk=6, muk=10, sigma=8, alpha=0.5, beta=0.7, p=2, eta=1e-3, epsilon=1e-4, k=0)`	mixequal.lagrange_augmented(funcs, args, cons_equal, cons_unequal, x_0)

import sympy as sp
import matplotlib.pyplot as plt
import optimtool as oo

f, x1, x2 = sp.symbols("f x1 x2")
f = (x1 - 2)**2 + (x2 - 1)**2
c1 = x1 - 2*x2
c2 = 0.25*x1**2 - x2**2 - 1
funcs = sp.Matrix([f])
cons_equal = sp.Matrix([c1])
cons_unequal = sp.Matrix([c2])
args = sp.Matrix([x1, x2])
x_0 = (0.5, 1)

f_list = []
title = ["penalty_quadratic", "penalty_L1", "lagrange_augmented"]
colorlist = ["maroon", "teal", "orange"]
_, _, f = oo.constrain.mixequal.penalty_quadratic(funcs, args, cons_equal, cons_unequal, x_0, False, True) # 第四个参数控制单个算法不显示迭代图，第五参数控制输出函数迭代值列表
f_list.append(f)
_, _, f = oo.constrain.mixequal.penalty_L1(funcs, args, cons_equal, cons_unequal, x_0, False, True)
f_list.append(f)
_, _, f = oo.constrain.mixequal.lagrange_augmented(funcs, args, cons_equal, cons_unequal, x_0, False, True)
f_list.append(f)

# 绘图
handle = []
for j, z in zip(colorlist, f_list):
    ln, = plt.plot([i for i in range(len(z))], z, c=j, marker='o', linestyle='dashed')
    handle.append(ln)
plt.xlabel("$Iteration \ times \ (k)$")
plt.ylabel("$Objective \ function \ value: \ f(x_k)$")
plt.legend(handle, title)
plt.title("Performance Comparison")
plt.show()

6. Lasso问题测试

from optimtool.example import Lasso

方法	函数参数	调用示例
梯度下降法	`gradient_descent(A, b, mu, args, x_0, draw=True, output_f=False, delta=10, alp=1e-3, epsilon=1e-2, k=0)`	Lasso.gradient_descent(A, b, mu, args, x_0,)
次梯度算法	`subgradient(A, b, mu, args, x_0, draw=True, output_f=False, alphak=2e-2, epsilon=1e-3, k=0)`	Lasso.subgradient(A, b, mu, args, x_0,)

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt
import optimtool as oo

import scipy.sparse as ss
f, A, b, mu = sp.symbols("f A b mu")
x = sp.symbols('x1:9')
m = 4
n = 8
u = (ss.rand(n, 1, 0.1)).toarray()
A = np.random.randn(m, n)
b = A.dot(u)
mu = 1e-2
args = sp.Matrix(x)
x_0 = tuple([1 for i in range(8)])

f_list = []
title = ["gradient_descent", "subgradient"]
colorlist = ["maroon", "teal"]
_, _, f = oo.example.Lasso.gradient_descent(A, b, mu, args, x_0, False, True, epsilon=1e-4)# 第四个参数控制单个算法不显示迭代图，第五参数控制输出函数迭代值列表
f_list.append(f)
_, _, f = oo.example.Lasso.subgradient(A, b, mu, args, x_0, False, True)
f_list.append(f)

# 绘图
handle = []
for j, z in zip(colorlist, f_list):
    ln, = plt.plot([i for i in range(len(z))], z, c=j, marker='o', linestyle='dashed')
    handle.append(ln)
plt.xlabel("$Iteration \ times \ (k)$")
plt.ylabel("$Objective \ function \ value: \ f(x_k)$")
plt.legend(handle, title)
plt.title("Performance Comparison")
plt.show()

7. WanYuan问题测试

from optimtool.example import WanYuan

方法	函数参数	调用示例
构造7个残差函数并采用高斯牛顿法	`gauss_newton(m, n, a, b, c, x3, y3, x_0, draw=False, eps=1e-10)`	WanYuan.gauss_newton(1, 2, 0.2, -1.4, 2.2, 2**(1/2), 0, (0, -1, -2.5, -0.5, 2.5, -0.05), draw=True)

问题描述：

给定直线方程的斜率（$m$）与截距（$n$），给定一元二次方程的二次项系数（$a$）、一次项系数（$b$）、常数（$c$），给定一个过定点的圆（$x_3$，$y_3$），要求这个过定点的圆与直线（$x_1$，$y_1$）和抛物线（$x_2$，$y_2$）相切的切点以及该圆的圆心（$x_0$，$y_0$）。

code：

# 导包
import sympy as sp
import matplotlib.pyplot as plt
import optimtool as oo

# 构造数据
m = 1
n = 2
a = 0.2
b = -1.4
c = 2.2
x3 = 2*(1/2)
y3 = 0
x_0 = (0, -1, -2.5, -0.5, 2.5, -0.05)

# 训练
oo.example.WanYuan.gauss_newton(1, 2, 0.2, -1.4, 2.2, 2**(1/2), 0, (0, -1, -2.5, -0.5, 2.5, -0.05), draw=True)

In v2.3.4, We call a method as follows:

import optimtool as oo
x1, x2, x3, x4 = sp.symbols("x1 x2 x3 x4")
f = (x1 - 1)**2 + (x2 - 1)**2 + (x3 - 1)**2 + (x1**2 + x2**2 + x3**2 + x4**2 - 0.25)**2
funcs = sp.Matrix([f])
args = sp.Matrix([x1, x2, x3, x4])
x_0 = (1, 2, 3, 4)
oo.unconstrain.gradient_descent.barzilar_borwein(funcs, args, x_0)

But in v2.3.5, We now call a method as follows: (It reduces the trouble of constructing data externally.)

import optimtool as oo
x1, x2, x3, x4 = sp.symbols("x1 x2 x3 x4") # Declare symbolic variables
f = (x1 - 1)**2 + (x2 - 1)**2 + (x3 - 1)**2 + (x1**2 + x2**2 + x3**2 + x4**2 - 0.25)**2
oo.unconstrain.gradient_descent.barzilar_borwein(f, [x1, x2, x3, x4], (1, 2, 3, 4)) # funcs, args, x_0
# funcs(args) can be list, tuple, sp.Matrix

Our function parameter input method is similar to matlab, and supports more methods than matlab.

Source code(tar.gz)
Source code(zip)

A Python step-by-step primer for Machine Learning and Optimization

early-ML Presentation General Machine Learning tutorials A Python step-by-step primer for Machine Learning and Optimization This github repository gat

8 Dec 1, 2022

Implementation of linesearch Optimization Algorithms in Python

Nonlinear Optimization Algorithms During my time as Scientific Assistant at the Karlsruhe Institute of Technology (Germany) I implemented various Opti

3 Dec 6, 2022

Microsoft contributing libraries, tools, recipes, sample codes and workshop contents for machine learning & deep learning.

366 Jan 3, 2023

A single Python file with some tools for visualizing machine learning in the terminal.

Minimize the Amount of Guided Packages

Is it necessary to reconstruct the matrix operation system of numpy and the symbolic algebra operation system of sympy in order to reduce the amount of dependent packets in the process of guilding packets.

opened by zzqwdwd 1

v1.5(Nov 10, 2022)

This version reduces the memory pressure caused by typing compared to v1.4.

import optimtool as oo
x1, x2, x3, x4 = sp.symbols("x1 x2 x3 x4") # Declare symbolic variables
f = (x1 - 1)**2 + (x2 - 1)**2 + (x3 - 1)**2 + (x1**2 + x2**2 + x3**2 + x4**2 - 0.25)**2
oo.unconstrain.gradient_descent.barzilar_borwein(f, [x1, x2, x3, x4], (1, 2, 3, 4)) # funcs, args, x_0

v1.4(Nov 8, 2022)

Use FuncArray, ArgArray, PointArray, IterPointType, OutputType in typing, and delete functions/ folder. I use many means to accelerate the method, I can't enumerate them here.

Tools for mathematical optimization region

Related tags

Overview

README.md

1. 无约束优化算法性能对比

2. 非线性最小二乘问题

3. 等式约束优化测试

4. 不等式约束优化测试

5. 混合等式约束测试

6. Lasso问题测试

7. WanYuan问题测试

You might also like...

A Python step-by-step primer for Machine Learning and Optimization

Implementation of linesearch Optimization Algorithms in Python

Microsoft contributing libraries, tools, recipes, sample codes and workshop contents for machine learning & deep learning.

A single Python file with some tools for visualizing machine learning in the terminal.

🔬 A curated list of awesome machine learning strategies & tools in financial market.

A Tools that help Data Scientists and ML engineers train and deploy ML models.

A collection of Scikit-Learn compatible time series transformers and tools.

Tools for Optuna, MLflow and the integration of both.

ClearML - Auto-Magical Suite of tools to streamline your ML workflow. Experiment Manager, MLOps and Data-Management

Comments

Minimize the Amount of Guided Packages

Releases(v1.5)

v1.5(Nov 10, 2022)

v1.4(Nov 8, 2022)

v1.3(Apr 25, 2022)

Owner

林景

XManager: A framework for managing machine learning experiments 🧑‍🔬

A data preprocessing package for time series data. Design for machine learning and deep learning.

Module for statistical learning, with a particular emphasis on time-dependent modelling

A Python implementation of GRAIL, a generic framework to learn compact time series representations.

Skoot is a lightweight python library of machine learning transformer classes that interact with scikit-learn and pandas.

A statistical library designed to fill the void in Python's time series analysis capabilities, including the equivalent of R's auto.arima function.

ML Optimizers from scratch using JAX

scikit-fem is a lightweight Python 3.7+ library for performing finite element assembly.

JMP is a Mixed Precision library for JAX.

Learn how to responsibly deliver value with ML.

Python based GBDT implementation

30 Days Of Machine Learning Using Pytorch

High performance, easy-to-use, and scalable machine learning (ML) package, including linear model (LR), factorization machines (FM), and field-aware factorization machines (FFM) for Python and CLI interface.

Simplify stop motion animation with machine learning.

Built on python (Mathematical straight fit line coordinates error predictor machine learning foundational model)

Multiple Linear Regression using the LinearRegression class from sklearn.linear_model library

Python package for machine learning for healthcare using a OMOP common data model

a distributed deep learning platform

Book Item Based Collaborative Filtering

Python package for causal inference using Bayesian structural time-series models.