STATS-你是一个ML专家，帮我解决这道问题，写出解决过程，不用到代码，纯手算的过程，英文。1.

你是一个ML专家，帮我解决这道问题，写出解决过程，不用到代码，纯手算的过程，英文。

Solution by Steps

step 1

To estimate the probability of Result

=1

for the given customer record, we first need to calculate the linear predictor

z

using the logistic regression coefficients:

z = \beta_0 + \beta_1 \cdot \text{Addr} + \beta_2 \cdot \text{Empl} + \beta_3 \cdot g + \beta_4 \cdot \text{Save}

Substituting the values:

z = 0.23 + 0.13 \cdot 7.5 + 0.22 \cdot 15 - 1.82 \cdot 1 + 0.0005 \cdot 800

step 2

Calculating each term:

z = 0.23 + 0.975 + 3.3 - 1.82 + 0.4 = 3.085

step 3

Now, we convert

z

to the estimated probability

\pi

using the logistic function:

\pi = \frac{1}{1 + e^{-z}} = \frac{1}{1 + e^{-3.085}} \approx 0.956

Answer

The estimated probability of Result

=1

for this customer is approximately

0.956

.

Key Concept

Logistic regression is used to model the probability of a binary outcome based on one or more predictor variables.

Explanation

The logistic function transforms the linear combination of predictors into a probability between 0 and 1, allowing us to interpret the results in the context of binary outcomes.

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Solution by Steps

step 1

To compute the overall fraction of correct predictions from the confusion matrix, we use the formula:

\text{Accuracy} = \frac{\text{True Positives} + \text{True Negatives}}{\text{Total Predictions}} = \frac{710 + 85}{1000} = 0.795

step 2

The false positive rate (FPR) is calculated as:

\text{FPR} = \frac{\text{False Positives}}{\text{False Positives} + \text{True Negatives}} = \frac{179}{179 + 710} \approx 0.201

step 3

The false negative rate (FNR) is calculated as:

\text{FNR} = \frac{\text{False Negatives}}{\text{False Negatives} + \text{True Positives}} = \frac{26}{26 + 85} \approx 0.234

Answer

The overall fraction of correct predictions is

0.795

, the false positive rate is approximately

0.201

, and the false negative rate is approximately

0.234

.

Key Concept

Confusion matrix metrics help evaluate the performance of a classification model.

Explanation

Accuracy, false positive rate, and false negative rate provide insights into how well the model predicts the binary outcomes, indicating areas for improvement.

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Solution by Steps

step 1

To lower the false negative rate, we can adjust the decision threshold for classifying a positive result. Instead of using the default threshold of

0.5

, we can lower it to increase sensitivity

step 2

This adjustment means that we will classify a result as positive if the predicted probability

\pi

is greater than a lower threshold

t

(where t < 0.5)

Answer

To lower the false negative rate, adjust the classifier's threshold to be less than

0.5

.

Key Concept

Adjusting the decision threshold can help balance sensitivity and specificity in classification tasks.

Explanation

Lowering the threshold increases the likelihood of classifying more instances as positive, which can reduce false negatives but may increase false positives.

---

Solution by Steps

step 1

For the new logistic regression model with the variable

h

, we write the log-odds equation:

\ln \left( \frac{\pi}{1 - \pi} \right) = \beta_0 + \beta_1 \cdot \text{Addr} + \beta_2 \cdot \text{Empl} + \beta_3 \cdot h + \beta_4 \cdot \text{Save} + \beta_5 \cdot h \cdot \text{Addr}

step 2

The maximum likelihood estimates (MLEs) of the parameters

\beta_0

to

\beta_5

can be derived from the logistic regression output. Assuming the coefficients from the previous model are used, we can denote them as follows:

\beta_0 = 0.23, \beta_1 = 0.13, \beta_2 = 0.22, \beta_3 = -1.82, \beta_4 = 0.0005, \beta_5 = -0.11

Answer

The MLEs of the parameters are:

\beta_0 = 0.23

,

\beta_1 = 0.13

,

\beta_2 = 0.22

,

\beta_3 = -1.82

,

\beta_4 = 0.0005

,

\beta_5 = -0.11

.

Key Concept

Maximum likelihood estimation is a method for estimating the parameters of a statistical model.

Explanation

MLE provides the parameter values that maximize the likelihood of observing the given data under the model, allowing for effective predictions.

你是一个ML专家，帮我解决这道问题，写出解决过程，不用到代码，纯手算的过程，英文。

Solution by Steps

step 1

The procedure of standardizing the gene features is correct because it ensures that each feature contributes equally to the model by having a mean of 0 and a standard deviation of 1. This is important in LASSO regression, which is sensitive to the scale of the input features

step 2

Using 5-fold cross-validation on the standardized features to select the tuning parameter

\lambda

is also appropriate. This method helps in estimating the prediction error by training the model on different subsets of the data and validating it on the remaining data

step 3

However, the response variable should not be standardized, as it is a continuous variable that we want to predict directly. Standardizing the response could lead to misleading interpretations of the model's predictions

step 4

To improve the procedure, I would suggest the following pseudo-code: 1. Split the dataset into training and validation sets. 2. For each

\lambda

in the grid: a. Standardize the gene features using only the training set. b. Fit the LASSO model on the standardized training set. c. Predict on the validation set using the original scale of the response. d. Calculate the prediction error. 3. Select the

\lambda

that minimizes the prediction error

Answer

The procedure is mostly correct, but the response variable should not be standardized. A better approach involves standardizing features only on the training set during cross-validation.

Key Concept

Standardization in regression models is crucial for ensuring that all features contribute equally, especially in LASSO regression.

Explanation

The procedure correctly standardizes features but should avoid standardizing the response variable to maintain interpretability. Adjusting the standardization process during cross-validation enhances model performance.

你是一个ML专家，帮我解决这道问题，写出解决过程，不用到代码，纯手算的过程，英文。

Solution by Steps

step 1

To find the missing numbers

z_1, z_2, z_3, z_4, z_5, z_6

, we need to calculate the deviance for each node using the formula:

\text{Deviance} = -2(n_1 \ln p_1 + n_0 \ln p_0)

where

n_1

and

n_0

are the number of observations for

y=1

and

y=0

respectively, and

p_1

and

p_0

are the probabilities for

y=1

and

y=0

step 2

For the root node (node 1), we have

n = 41034

,

yval = 0

(which means

p_1 = 0.91399820

and

p_0 = 0.0860018

). Thus, we can calculate the deviance:

\text{Deviance} = -2(41034 \cdot \ln(0.91399820) + (41034 - n_1) \cdot \ln(0.0860018)

We need to find

n_1

for node 1

step 3

For node 2, we have

n = 41034

,

yval = 0

(thus

p_1 = 0.91399820

and

p_0 = 0.0860018

). The deviance is already given as

24061.40

. We can use this to find

z_1

and

z_2

step 4

For node 3, we have

n = 2606

,

yval = 1

(thus

p_1 = 0.3779739

and

p_0 = 0.6220261

). We can calculate the deviance:

\text{Deviance} = -2(2606 \cdot \ln(0.3779739) + (2606 - n_1) \cdot \ln(0.6220261)

We need to find

n_1

for node 3

step 5

For the false positive rate, we calculate it as:

\text{False Positive Rate} = \frac{\text{False Positives}}{\text{False Positives} + \text{True Negatives}} = \frac{179}{179 + 710}

This gives us the in-sample false positive rate

Answer

The missing numbers

z_1, z_2, z_3, z_4, z_5, z_6

can be calculated using the deviance formula, and the in-sample false positive rate is approximately

0.201

.

Key Concept

Deviance is a measure of the goodness of fit of a model, and the false positive rate indicates the proportion of incorrect positive predictions.

Explanation

The calculations for deviance help identify the missing values in the classification tree, while the false positive rate provides insight into the model's performance.

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