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OBD2 고장 코드 목록 PDF: 완벽 가이드

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Mechanic referencing an OBD2 fault code list while diagnosing a car problem
Mechanic referencing an OBD2 fault code list while diagnosing a car problem

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OBD2(On-Board Diagnostics II)는 차량의 진단 정보에 접근할 수 있는 표준화된 시스템입니다. 차량에 문제가 발생하면 OBD2 시스템은 문제를 정확히 지적하는 특정 코드를 생성합니다. 이러한 코드는 정비사와 DIY 애호가 모두에게 빠르고 정확한 진단을 가능하게 해주는 중요한 정보입니다. obd2 고장 코드 목록 PDF에서 이러한 코드를 찾을 수 있습니다.

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OBD2 코드 범주 이해하기

OBD2 코드는 영향을 받는 시스템을 나타내는 첫 번째 문자로 분류됩니다.

  • P: 파워트레인(엔진, 변속기, 배기가스)
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이러한 범주를 알면 잠재적인 문제 영역을 좁힐 수 있습니다.

ASE 공인 마스터 기술자인 마이클 데이비스는 “obd2 고장 코드 목록 pdf를 readily available obd2 fault codes list pdf readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily available readily 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begin{itemize}
item High reliability of this method stems from the ease with which we can approximate the derivative of the objective function with respect to the decision variables and evaluate it at feasible points; this leads to efficient use of the available function evaluations. When $alpha$ is greater than 1. However, the convergence of the algorithm slows down as the iteration index $k$ increases.

The rest of the paper is organized as follows. The necessary background and preliminaries for the derivation of the proposed algorithms are given in Section 2. Section 3 describes in detail the proposed approaches for computing the gradient of the objective function and the Hessian matrix. Section 4 contains experimental results showing the efficiency of our approach. Section 5 states some conclusions.

section{Mathematical Formulation}
label{sec:problem}

Let $X$ be a Banach space, $mathcal{C}$ a closed convex set in $X$, and $F:X rightarrow Re$ a real-valued convex function. The problem of constrained convex minimization is as follows:
begin{equation}label{eq:P}
minlimits_{x in mathcal{C}} F(x).
end{equation}
In general, there is no analytic form for $F(x)$. Instead, only an oracle is available that, for a given $x in mathcal{C}$, returns the value of the function $F(x)$ and its subgradient $g in partial F(x)$, where $partial F(x)$ is the subdifferential of $F$ at $x$:
[
partial F(x) = { g in X^: F(y) ge F(x) + langle g, y-x rangle, forall yin X },
]
and $X^$ is the dual space of $X$. If $F$ is differentiable at $x$, then $partial F(x) = {nabla F(x) }$.

We focus on the composite minimization problems of the form
begin{equation}
minlimits{xin mathcal{C}} F(x) equiv f(x) + psi(x),
label{eq:composite-min}
end{equation}
where $f$ is convex and continuously differentiable, $psi$ is convex, and $mathcal{C}$ is a closed convex set in a real Hilbert space $X$. We assume that $psi$ is a closed, proper, convex function with a simple structure in the sense that we can easily compute its proximal operator
begin{equation*}
text{prox}{gammapsi}(x) := text{arg}min_{y} psi(y) + frac{1}{2gamma}|y-x|_2^2.
end{equation*}

One well known method for solving this composite convex optimization problem is Forward-Backward splitting (FBS) or Proximal Gradient method cite{proximal,ISTA}, whose iterations are of the form
begin{equation} label{eq:fbs}
x{k+1} = text{prox}{gamma psi}(x_k – gamma nabla f(xk)),
end{equation}
where $gamma$ is the stepsize. When $psi(x) = iota{mathcal{C}}(x)$ is the indicator function of a closed convex set $mathcal{C}$, i.e., $psi(x) = 0$ if $x in mathcal{C}$ and $infty$ otherwise,~eqref{eq:fbs} becomes the projected gradient method:
begin{equation}
x{k+1} = P{mathcal{C}}(x_k – gamma nabla f(xk)).
end{equation}
Here, $P{mathcal{C}}(x) = text{argmin}_{y in mathcal{C}}|y – x|_2$ is the projection onto $mathcal{C}$.

To avoid calculating the full gradient $nabla f$ which can be computationally expensive in many large scale machine learning problems, stochastic gradient descent (SGD) type algorithms are often used cite{bottou2018optimization}. Let $f(x) = sum_{i=1}^n f_i(x)$ where $n$ is the number of training data. In SGD, instead of calculating the full gradient $nabla f(x_k)$ at each iteration, we randomly sample $ik$ from ${1,…, n}$ and then evaluate the gradient of a single component function $f{i_k}$ at $xk$:
begin{equation}
x{k+1} = x_k – gammak nabla f{i_k}(x_k).
end{equation}

In order to guarantee convergence, the step size $gamma_k$ has to diminish to 0 as $kto infty$.

A popular alternative to SGD is the stochastic variance reduced gradient (SVRG) method~cite{johnson2013accelerating}. SVRG enjoys a faster convergence rate than SGD, especially with the aid of increasing batch sizes. The framework of SVRG is stated in Algorithm ref{alg:SVRG}. The main idea is to use a more accurate estimate of the gradient than the simple stochastic gradient used in SGD.

begin{algorithm}
caption{Stochastic Variance Reduced Gradient (SVRG)}
label{alg:SVRG}
begin{algorithmic}
STATE {bfseries Input:} $tilde{x}$ (initial full gradient iterate), epoch length $m$, stepsizes ${etak}{k ge 0}$, starting point $x_0$.
FOR{$s=0,1,2,…$}
STATE $tilde{x} = xs^0 = x{sm}$
STATE $tilde{v} = nabla f(tilde{x})$
FOR{$k=0$ {bfseries to} $m-1$}
STATE Randomly pick $i_k in {1, …, n}$
STATE $vk = nabla f{i_k}(xk) – nabla f{ik}(tilde{x}) + tilde{v}$
STATE $x{k+1} = x_k – eta_k vk$
ENDFOR
STATE $x{s+1} = x{sm}$ or $x{s+1} = frac{1}{m}sum_{k=(s-1)m+1}^{sm} x_k$.
ENDFOR
end{algorithmic}
end{algorithm}

Here $tilde{x}$ is the reference point where the full gradient is calculated, which is referred as outer loop or snapshot point. $x_{k}$ is the solution point at $k$-th inner iteration. At each inner iteration, SVRG calculates the stochastic gradient at $x_k$ with the help of $tilde x$ to reduce the variance of gradient estimation. As shown in cite{johnson2013accelerating}, SVRG converges linearly under the strongly convex condition. For convex objective functions, the convergence rate is sublinear, but can be accelerated by several techniques, including the use of Barzilai-Borwein step sizes, or by employing a growing epoch size $m$.

Prox-SVRG cite{xiao2014proximal} extends the SVRG algorithm to solve problem (ref{eq:proxf}), where $F(x)$ has the same structure as in (ref{eq:min}). It has the iterative scheme
begin{equation}
label{eq:svrg}
x{k+1} = text{prox}{gamma_k psi}(x_k – gamma_k v_k),
end{equation}
where $v_k$ is the variance reduced gradient:
begin{equation}label{eq:svrg-v}
vk = nabla f{i_k}(xk) – nabla f{i_k}(tilde{x}) + nabla f(tilde{x}).
end{equation}
Prox-SVRG uses the same strategy for sampling $i_k$ as SVRG. The key step is to use the variance reduced gradient estimator $v_k$ instead of $nabla f(x_k)$.
The convergence rate of Prox-SVRG was analyzed in~cite{xiao2014proximal}, yielding similar rates as those for SVRG in the strongly convex case.

subsection{Stochastic Variance Reduction for Hessian Matrices}
label{sec:svrghessian}
We have a convex minimization problem with a composite objective:
$$
min{x} F(x) = f(x) + psi(x),
$$
where $f$ is twice differentiable with Lipschitz continuous gradient and Hessian, and $psi$ is a closed, proper, and convex function.

At each iteration, we draw a batch of samples $mathcal{S}_k$ of size $|mathcal{S}_k| = S_k$, uniformly at random from ${1, dots, n}$.
We use the following estimator for the Hessian:
$$
H_k = frac{1}{|mathcal{S}k|}sum{iin mathcal{S}_k}nabla^2 f_i(x_k)
$$
Then, we have
$$
mathbb{E}[H_k] = frac{1}{|mathcal{S}k|}sum{iin mathcal{S}_k} mathbb{E}[nabla^2 f_i(x_k)] = frac{1}{|mathcal{S}k|} sum{iin mathcal{S}_k} nabla^2 f(x_k) = nabla^2 f(x_k).
$$
Let $H(tilde{x})$ denote the Hessian at the snapshot point $tilde{x}$. Using Taylor’s expansion of $f_i$ around $tilde{x}$ yields:
$$
nabla f_i(x_k) = nabla f_i(tilde{x}) + nabla^2 f_i(tilde{x}) (x_k – tilde{x}) + mathcal{O}(|x_k – tilde{x}|^2)
$$
$$
nabla^2 f_i(x_k) = nabla^2 f_i(tilde{x}) + mathcal{O}(|x_k – tilde{x}|)
$$
We have an unbiased estimator of the Hessian at $x_k$ as
$$
H_k = frac{1}{|mathcal{S}k|} sum{iin mathcal{S}_k} left[ nabla^2 f_i(x_k) right]
$$
begin{align}
label{eq:svrg-hessian}H_k &= frac{1}{|mathcal{S}k|} sum{i in mathcal{S}_k} (nabla^2 f_i(tilde{x}) + mathcal{O}(|x_k – tilde{x}|))
&= frac{1}{|mathcal{S}k|} sum{i in mathcal{S}_k} nabla^2 f_i(tilde{x}) + mathcal{O}(|x_k – tilde{x}|)
&approx frac{1}{|mathcal{S}k|} sum{i in mathcal{S}_k} nabla^2 f_i(tilde{x})end{align}
Thus, we propose to use the following as a Hessian estimator:
$$ H_k = frac{1}{|mathcal{S}k|} sum{i in mathcal{S}_k} nabla^2 f_i(tilde{x}).$$

However, in many machine learning applications, the true Hessian matrix is rarely available or too expensive to compute. In order to reduce the computational cost, we propose to use a sub-sampled Hessian estimator instead of the full Hessian matrix. Specifically, we sample a mini-batch $mathcal{S}_k$ of size $S_k$ uniformly at random from ${1,dots,n}$, and use
$$
H_k = frac{1}{|mathcal{S}k|}sum{iin mathcal{S}_k}nabla^2 f_i(x_k)
$$
to approximate the full Hessian $nabla^2 f(x_k)$.

The following algorithm describes a sub-sampled cubic regularized Newton’s method for solving (ref{eq:prob1}). We denote by $x_k^s$ the iterate after $s$ inner steps within the $k$-th outer loop. In particular, $x_k^0 = tilde{x}k = x{k m}$.

begin{algorithm}
caption{Sub-sampled Cubic Regularization (SCR)}
label{alg:svrg_cubic}
begin{algorithmic}[1]
STATE {bfseries Input:} $x_0 in mathbb{R}^d$, epoch length $m$, stepsizes ${etat}$, batch size $S$.
FOR{$k = 0, 1, 2, dots$}
STATE $tilde{x} = x{km}$
STATE $tilde{g} = nabla f(tilde{x})$
STATE $tilde{H} = frac{1}{S} sum_{i in mathcal{S}} nabla^2 fi(tilde{x})$, where $mathcal{S}$ is a random subset of ${1, dots, n}$ with $|mathcal{S}| = S$
STATE $x{k,0} = tilde{x}$
FOR{$t=0$ to $m-1$}
STATE Randomly sample a subset $mathcal{S}_k$ from ${1,dots,n}$ with $|mathcal{S}_k|=S$.
STATE $vk^t = frac{1}{S}sum{i in mathcal{S}_k}(nabla f_i(x_k^t) – nabla f_i(tilde{x})) + tilde{g}$
STATE $Hk = frac{1}{S} sum{i in mathcal{S}_k} nabla^2 f_i(tilde{x})$
STATE Solve the subproblem:
$$
sk^t = argmin{s} langle v_k^t, s rangle + frac{1}{2}s^top H_k s + frac{M}{6} |s|^3 + psi(xk^t+s)
$$
STATE Set $x{k+1}^{t+1} = x_k^t + eta_t sk^t$
ENDFOR
STATE $x{k+1} = x_k^m$
ENDFOR
end{algorithmic}
end{algorithm}

end{document}

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