数学代写|凸优化作业代写Convex Optimization代考|CPD131

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数学代写|凸优化作业代写Convex Optimization代考|Detection probability matrix

For the randomized detector defined by the matrix $T$, we define the detection probability matrix as $D=T P$. We have
$$D_{i j}=(T P){i j}=\operatorname{prob}(\hat{\theta}=i \mid \theta=j)$$ so $D{i j}$ is the probability of guessing $\hat{\theta}=i$, when in fact $\theta=j$. The $m \times m$ detection probability matrix $D$ characterizes the performance of the randomized detector defined by $T$. The diagonal entry $D_{i i}$ is the probability of guessing $\hat{\theta}=i$ when $\theta=i$, i.e., the probability of correctly detecting that $\theta=i$. The off-diagonal entry $D_{i j}$ (with $i \neq j$ ) is the probability of mistaking $\theta=i$ for $\theta=j$, i.e., the probability that our guess is $\hat{\theta}=i$, when in fact $\theta=j$. If $D=I$, the detector is perfect: no matter what the parameter $\theta$ is, we correctly guess $\hat{\theta}=\theta$.

The diagonal entries of $D$, arranged in a vector, are called the detection probabilities, and denoted $P^{\mathrm{d}}$ :
$$P_i^{\mathrm{d}}=D_{i i}=\operatorname{prob}(\hat{\theta}=i \mid \theta=i)$$
The error probabilities are the complements, and are denoted $P^{\mathrm{e}}$ :
$$P_i^e=1-D_{i i}=\operatorname{prob}(\hat{\theta} \neq i \mid \theta=i)$$
Since the columns of the detection probability matrix $D$ add up to one, we can express the error probabilities as
$$P_i^{\mathrm{e}}=\sum_{j \neq i} D_{j i}$$

数学代写|凸优化作业代写Convex Optimization代考|Bias, mean-square error, and other quantities

In this section we assume that the ordering of the values of $\theta$ have some significance, i.e., that the value $\theta=i$ can be interpreted as a larger value of the parameter than $\theta=j$, when $i>j$. This might be the case, for example, when $\theta=i$ corresponds to the hypothesis that $i$ events have occurred. Here we may be interested in quantities such as
$$\operatorname{prob}(\hat{\theta}>\theta \mid \theta=i)$$
which is the probability that we overestimate $\theta$ when $\theta=i$. This is an affine function of $D$ :
$$\operatorname{prob}(\hat{\theta}>\theta \mid \theta=i)=\sum_{j>i} D_{j i}$$ so a maximum allowable value for this probability can be expressed as a linear inequality on $D$ (hence, $T$ ). As another example, the probability of misclassifying $\theta$ by more than one, when $\theta=i$,
$$\operatorname{prob}(|\hat{\theta}-\theta|>1 \mid \theta=i)=\sum_{|j-i|>1} D_{j i}$$
is also a linear function of $D$.
We now suppose that the parameters have values $\left{\theta_1, \ldots, \theta_m\right} \subseteq \mathbf{R}$. The estimation or detection (parameter) error is then given by $\hat{\theta}-\theta$, and a number of quantities of interest are given by linear functions of $D$. Examples include:

• Bias. The bias of the detector, when $\theta=\theta_i$, is given by the linear function
$$\underset{i}{\mathbf{E}}(\hat{\theta}-\theta)=\sum_{j=1}^m\left(\theta_j-\theta_i\right) D_{j i}$$
where the subscript on $\mathbf{E}$ means the expectation is with respect to the distribution of the hypothesis $\theta=\theta_i$.
• Mean square error. The mean square error of the detector, when $\theta=\theta_i$, is given by the linear function
• Average absolute error. The average absolute error of the detector, when $\theta=\theta_i$, is given by the linear function
$$\underset{i}{\mathbf{E}}|\hat{\theta}-\theta|=\sum_{j=1}^m\left|\theta_j-\theta_i\right| D_{j i}$$

数学代写|凸优化作业代写Convex Optimization代考|Detection probability matrix

$$D_{i j}=(T P) i j=\operatorname{prob}(\hat{\theta}=i \mid \theta=j)$$

$$P_i^{\mathrm{d}}=D_{i i}=\operatorname{prob}(\hat{\theta}=i \mid \theta=i)$$

$$P_i^e=1-D_{i i}=\operatorname{prob}(\hat{\theta} \neq i \mid \theta=i)$$

$$P_i^{\mathrm{e}}=\sum_{j \neq i} D_{j i}$$

数学代写|凸优化作业代写Convex Optimization代考|Bias, mean-square error, and other quantities

$$\operatorname{prob}(\hat{\theta}>\theta \mid \theta=i)$$

$$\operatorname{prob}(\hat{\theta}>\theta \mid \theta=i)=\sum_{j>i} D_{j i}$$

$$\operatorname{prob}(|\hat{\theta}-\theta|>1 \mid \theta=i)=\sum_{|j-i|>1} D_{j i}$$

Veft{{theta_1, \dots, Itheta_m Iright $}$ Isubseteq $\backslash m a t h b f{R}$
.然后由下式给出估计或检测（参数）误差 $\hat{\theta}-\theta$ ，并且

• 偏见。检测器的偏置，当 $\theta=\theta_i$, 由线性函数给出
下标在哪里 $\mathbf{E}$ 表示期望与假设的分布有关 $\theta=\theta_i$.
• 均方误差。检测器的均方误差，当 $\theta=\theta_i$, 由线性 函数给出
• 平均绝对误差。检测器的平均绝对误差, 当 $\theta=\theta_i$, 由线性函数给出

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