Fahime Sokhangou You may use the following procedures to determine the studentized residual for each observation in Locally Weighted Regression (LOWESS) using a second-order polynomial:

1. Using the LOWESS technique, compute the anticipated values for each observation.

2. Subtract the projected values from the actual values to get the residuals for each observation.

3. Calculate the hat matrix for each observation, which is the hat matrix's diagonal entry. For each observation, the hat matrix is a 10x10 diagonal matrix.

4. Divide the relevant diagonal element of the hat matrix by the total number of observations to get the leverage value for each observation.

5. Calculate the studentized residual for each observation by dividing the residual by the square root of the residual mean square multiplied by the number of observations (1-leverage value)

6. Finally, you may check for outliers in your data by comparing the estimated studentized residuals to the critical values of the t-distribution.

The studentized residual formula is (residual / (residual mean square * (1- leverage value))1/2).

It's worth noting that in the case of locally weighted regression, the span parameter specifies how much weight is given to data near the point of interest.

The tricube function is also used to weight the observations; it is a weighing function that calculates the weight of each observation.

Qamar Ul Islam

Thanks a lot.

I exactly did same as what you explained until step 4.

I have 26 points and as a results 26 hat matrix.

I do not know what is the relevant diagonal element in the step 4?

''Divide the relevant diagonal element of the hat matrix by the total number of observations to get the leverage value for each observation.''

Can you please explain this part?

I appreciate your time.

Qamar Ul Islam for example the hat matrix for the first observation is like the file that I have attached.

Fahime Sokhangou The relevant diagonal element of the hat matrix is the diagonal element of the hat matrix that corresponds to a specific observation in step 4. The hat matrix, also known as the projection matrix, is used in a linear regression model to project the observed data into the space spanned by the predictors. The leverage of a given observation is represented by each diagonal member of the hat matrix, which is a measure of how much the observation deviates from the overall pattern of the data.

Divide the appropriate diagonal element of the hat matrix by the total number of observations to obtain the leverage value for a single observation. This yields a number between 0 and 1, indicating the fraction of overall variance in the response variable explained by that particular observation.

The leverage is helpful for detecting outliers, high-leverage points, and influential observations, as well as for identifying outliers, high-leverage points, and influential observations.

Qamar Ul Islam Thanks a lot.

So, how to select the appropriate diagonal element?

Imagine in the hat matrix that I have (10*10) for each observation, which element is hii?

Fahime Sokhangou In linear regression, the "hat matrix" (also known as the projection matrix or the impact matrix) is a matrix that reflects the linear transformation that maps the response variable to its predicted values. The diagonal components of the hat matrix (denoted as h ii) are known as leverage values, which are measurements of each observation's effect on the regression model.

To get the relevant diagonal element in the hat matrix, use the row and column indices of the observation in question. If you have a 10x10 hat matrix and wish to calculate the leverage value of the i-th observation, for example, you may extract the i-th diagonal element as follows:

hii = hat_matrix[i,i]

Keep in mind that leverage values are not always simple to comprehend, and they are more commonly used in conjunction with other impact metrics, such as the Cook's distance or the DFBETA.

Qamar Ul Islam Thanks a lot, Finally I got what do you mean by the releavant observation.

For example, if I am predicting the observation number 1, I should determine the hii which is the first diagonal element in Hat matrix.

I appreciate your help.

Fahime Sokhangou Yes, that's right. The relevant value for predicting a given observation is the diagonal element of the hat matrix that corresponds to that observation. If you are forecasting the first observation, the relevant value in your case would be the first diagonal member of the hat matrix.