How to find variance percentage given VIF

 ## **Step-by-Step Solution**

### **1. Understanding VIF Formula**

The Variance Inflation Factor is:

\[

\text{VIF} = \frac{\text{Actual variance of coefficient}}{\text{Variance with no multicollinearity}}

\]

Given: **VIF = 1.8**

### **2. Interpret the VIF Value**

\[

1.8 = \frac{\text{Actual variance}}{\text{Variance with no multicollinearity}}

\]

This means the actual variance is **1.8 times** what it would be with no multicollinearity.

### **3. Calculate Percentage Increase**

If variance with no multicollinearity = 1 (base), then:

- Actual variance = 1.8

- **Increase** = 1.8 - 1 = 0.8

- **Percentage increase** = \( \frac{0.8}{1} \times 100\% = 80\% \)

---

## **Final Answer**

\[

\boxed{80}

\]

The variance of the coefficient is **80% greater** than what it would be if there was no multicollinearity.

---

### **Verification**

- VIF = 1.0 → 0% increase (no multicollinearity)

- VIF = 2.0 → 100% increase (variance doubles)

- VIF = 1.8 → 80% increase ✓

This makes intuitive sense: moderate multicollinearity (VIF = 1.8) inflates the variance by 80% compared to the ideal case.

What is Variable Inflation factor?

## **Variance Inflation Factor (VIF)**

The **Variance Inflation Factor (VIF)** measures how much the variance of a regression coefficient is inflated due to multicollinearity in the model.

---

### **Formula**

For predictor \( X_k \):

\[

\text{VIF}_k = \frac{1}{1 - R_k^2}

\]

where \( R_k^2 \) is the R-squared value from regressing \( X_k \) on all other predictors.

---

### **Interpretation**

- **VIF = 1**: No multicollinearity

- **1 < VIF ≤ 5**: Moderate correlation (usually acceptable)

- **VIF > 5 to 10**: High multicollinearity (may be problematic)

- **VIF > 10**: Severe multicollinearity (coefficient estimates are unstable)

---

## **How VIF is Helpful**

1. **Detects Multicollinearity**

   - Identifies when predictors are highly correlated with each other

   - Helps understand which variables contribute to collinearity

2. **Assesses Regression Coefficient Stability**

   - High VIF → large standard errors → unreliable coefficient estimates

   - Helps decide if some variables should be removed or combined

3. **Guides Model Improvement**

   - Suggests when to:

     - Remove redundant variables

     - Combine correlated variables (e.g., using PCA)

     - Use regularization (Ridge regression)

4. **Better Model Interpretation**

   - With lower multicollinearity, coefficient interpretations are more reliable

   - Each predictor's effect can be isolated more clearly

---

### **Example Usage**

If you have predictors: House Size, Number of Rooms, Number of Bathrooms

- Regress "Number of Rooms" on "House Size" and "Number of Bathrooms"

- High \( R^2 \) → High VIF → these variables contain overlapping information

- Solution: Maybe use only "House Size" and one other, or create a composite feature

---

**Bottom line**: VIF helps build more robust, interpretable models by identifying and addressing multicollinearity issues.

 

What is Q-Q plot and their benefits

A Q-Q (quantile-quantile) plot compares the quantiles of two distributions.

If the two distributions are identical (or very close), the points on the Q-Q plot will fall approximately along the 45° straight line 

A **Q-Q plot** (quantile-quantile plot) is a graphical tool used to compare two probability distributions by plotting their quantiles against each other.

---

## **How it works**

- One distribution’s quantiles are on the x-axis, the other’s on the y-axis.

- If the two distributions are similar, the points will fall roughly along the **line \(y = x\)** (the 45° diagonal).

- Deviations from this line indicate how the distributions differ in shape, spread, or tails.

---

## **Types of Q-Q plots**

1. **Two-sample Q-Q plot**: Compare two empirical datasets.

2. **Theoretical Q-Q plot**: Compare sample data to a theoretical distribution (e.g., normal Q-Q plot to check normality).

---

## **Benefits of Q-Q plots**

1. **Visual check for distribution similarity**  

   - Quickly see if two datasets come from the same distribution family.

2. **Assess normality**  

   - Common use: Normal Q-Q plot to check if data is approximately normally distributed.

3. **Identify tails behavior**  

   - Points deviating upward at the top → right tail of sample is heavier than theoretical.  

   - Points deviating downward at the top → right tail is lighter.

4. **Detect skewness**  

   - A curved pattern suggests skew.

5. **Spot outliers**  

   - Points far off the line may be outliers.

6. **Compare location and scale differences**  

   - If points lie on a straight line with slope ≠ 1 → scale difference.  

   - If intercept ≠ 0 → location shift.

---

## **Example interpretation**

- **Straight diagonal line**: Distributions are the same.

- **Straight line with slope > 1**: Sample has greater variance.

- **S-shaped curve**: Tails differ (one distribution has heavier or lighter tails).

- **Concave up**: Sample distribution is right-skewed relative to theoretical.

Minikube: basic minikube and kubctl commands

Minikube: kubectl to create deployment 

# start minikube 

minikube start

# view minikube dashboard 

minikube dashboard

#get all the deployments 

kubectl get deployments

kubectl get deployments -n <namespace name>

#View the pods:

kubectl get pods

kubectl get pods -n <namespace name>

#View cluster events:

kubectl get events

kubectl get events -n <namespace name>

# View the kubectl configuration

kubectl config view

kubectl config view -n <namespace name>

kubectl logs <pod name>

kubectl logs <pod name> -n dev

# get kubectl services 

kubectl get services

kubectl get services

# list the addons in minikube 

minikube addons list

#enable a specific add on ( in this case, enabling metrics-server) 

minikube addons enable <metric name>

#for e.g. To enable ingress 

minikube addons enable ingress

Minikube : creating kubernetes cluster

Kubernetes coordinates a highly available cluster of computers that are connected to work as a single unit. The abstractions in Kubernetes allow you to deploy containerized applications to a cluster without tying them specifically to individual machines. To make use of this new model of deployment, applications need to be packaged in a way that decouples them from individual hosts: they need to be containerized. Containerized applications are more flexible and available than in past deployment models, where applications were installed directly onto specific machines as packages deeply integrated into the host. Kubernetes automates the distribution and scheduling of application containers across a cluster in a more efficient way. Kubernetes is an open-source platform and is production-ready.

A Kubernetes cluster consists of two types of resources:

The Control Plane coordinates the cluster

Nodes are the workers that run applications

The Control Plane is responsible for managing the cluster. The Control Plane coordinates all activities in your cluster, such as scheduling applications, maintaining applications' desired state, scaling applications, and rolling out new updates.

A node is a VM or a physical computer that serves as a worker machine in a Kubernetes cluster. 

Each node has a Kubelet, which is an agent for managing the node and communicating with the Kubernetes control plane. The node should also have tools for handling container operations, such as containerd or CRI-O. A Kubernetes cluster that handles production traffic should have a minimum of three nodes because if one node goes down, both an etcd member and a control plane instance are lost, and redundancy is compromised. You can mitigate this risk by adding more control plane nodes.

When you deploy applications on Kubernetes, you tell the control plane to start the application containers. The control plane schedules the containers to run on the cluster's nodes. Node-level components, such as the kubelet, communicate with the control plane using the Kubernetes API, which the control plane exposes. End users can also use the Kubernetes API directly to interact with the cluster.

A Kubernetes cluster can be deployed on either physical or virtual machines. To get started with Kubernetes development, you can use Minikube. Minikube is a lightweight Kubernetes implementation that creates a VM on your local machine and deploys a simple cluster containing only one node. Minikube is available for Linux, macOS, and Windows systems. The Minikube CLI provides basic bootstrapping operations for working with your cluster, including start, stop, status, and delete.