SERIE DE TIEMPO-EXPLICACIÓN Y EJEMPLO (CONTIENE LA MAYORIA DE LOS COMPONENTES) ANALISIS DE TENDENCIA

Understanding Time Series Analysis

Introduction to Time Series

• A time series is defined as a collection of observations recorded at successive intervals over time, which can be weekly, monthly, quarterly, or annually.

• While graphical representation helps in observing trends, it may not always reveal the unique characteristics of each series.

Components of Time Series

• There are four main components in a time series: secular trend, cyclical variation, seasonal variation, and irregular variation (also known as random variation).

Secular Trend

• The secular trend indicates a long-term increase or decrease in a variable's value; for example, the consistent rise in living costs reflected in consumer price indices.

Cyclical Variation

• Cyclical variations represent fluctuations that occur over longer periods; an example is economic cycles where business activity peaks above or dips below the trend line.

Seasonal Variation

• Seasonal variations show regular patterns within a year that repeat annually; these are useful for forecasting future trends. For instance, sales often peak during specific quarters.

Irregular Variation

• Irregular variations describe unpredictable changes that occur randomly and do not follow any discernible pattern.

Analyzing Time Series Data

• Most time series contain multiple components simultaneously. Understanding this allows for comprehensive analysis by breaking down total variation into its constituent parts.

Practical Application: Forecasting Sales

• The analysis of time series is crucial for identifying patterns in statistical information at regular intervals to project future outcomes.

Steps to Analyze Sales Data

1. Desseasonalization: Remove seasonal effects from the data.

2. Trend Line Development: Establish a baseline trend from historical data.

3. Cyclical Variation Identification: Assess fluctuations around the established trend line based on historical sales data from 1991 to 1995.

Desseasonalization Explained

• Desseasonalization involves calculating seasonal indices to adjust the original data and remove seasonal effects for clearer insights into underlying trends.

Understanding Seasonality

• Seasonality refers to predictable patterns occurring at regular intervals (e.g., increased ice cream sales during summer months).

Process of Desseasonalization

• A desseasonalized series reflects behavior without seasonal spikes using mathematical methods like moving averages to smooth out fluctuations.

Moving Average Calculation

• To calculate moving averages:

• Gather quarterly sales data and compute totals over four quarters.

• Divide these totals by four to find average values per quarter.

Centering Moving Averages

Calculating Seasonal Indices and Trend Lines

Understanding the Calculation of Seasonal Indices

• The percentage of the actual value relative to the moving average can be calculated, although this step may be omitted when centering data. For a weekly moving average, it centers on the fourth day.

• To find this percentage, divide the centered value (9 in this case) by 15.875 and multiply by 100, resulting in 56.7%. This process is repeated for all other values.

• A modified sum must be calculated after removing the highest and lowest values from each quarter per year; for instance, in Q1, remove 89.5 (min) and 92.9 (max).

• The remaining values are summed to obtain a modified sum which is then divided by two to get a modified mean—this represents the unadjusted seasonal index.

• The adjustment factor is computed by summing all modified means and dividing 400 by this total (397.64), yielding an adjustment factor of approximately 1.006.

Desseasonalizing Time Series Data

• Using obtained seasonal indices, desseasonalized sales are calculated by dividing actual sales figures by their respective seasonal indices; results include values like 16.83 for Q1 and others accordingly.

• After completing desseasonalization, we proceed to develop a trend line using these desseasonalized sales data with methods such as semi-averages or least squares.

Methods for Finding Trend Line Equations

• Various methods exist for determining trend lines; semi-average method provides less precision than least squares but is simpler depending on user preference and organizational needs.

• To apply the semi-average method: split periods into two equal parts (e.g., periods 1–10 and 11–20), calculate central X values for both sides—5.5 for one side and 15.5 for another.

• Semi-means are derived from averaging these central values; first part yields approximately 16.321 while second gives about 19.792.

Calculating Trend Line Equation

• The trend line equation takes form y = a + b cdot x. Values a and b are determined through substitution into equations based on earlier calculations.

• By eliminating variables during calculations, we find that a_1approx0.369. Substituting back gives us a_0 =14.410.

Finalizing Sales Forecasting

• The final trend line equation becomes y =14.410 +0.346x. This allows forecasting future sales up to period 20 plus four additional forecasted periods for year-end predictions in 1996.

• With these forecasts established, we move onto calculating cyclical variation around the trend line using previously computed desseasonalized sales data.

Analyzing Cyclical Variation

Analysis of Time Series Components

Understanding Seasonal and Cyclical Variations

• The component of a time series that fluctuates above and below the linear trend over periods longer than one year is identified as cyclical variation. It is important to note that seasonal variation completes a full cycle within each year and does not affect one year more than another.

• Two methods measure cyclical variation:

• The first method expresses the variation as a percentage of the trend, known as the "percentage of trend."

• The second method, called relative cyclical residual, calculates the variation as a percentage deviation from the trend.

Forecasting Sales with Seasonal Adjustments

• To create an accurate forecast, it is necessary to seasonally adjust predicted sales data. This involves multiplying deserialized sales data by previously calculated seasonal indices to obtain seasonally adjusted forecasts.

• An acceptable forecast considers cyclical variations and typically has around a 10% variance. A comprehensive analysis of time series aims to explain secular trends, cyclical variations, and seasonal variations while identifying irregular variations.

Steps for Effective Time Series Analysis

• The correct approach for analyzing all components of a time series includes:

• First, deseasonalizing the time series.

• Next, finding the trend line.

• Then calculating variations around this trend line.

• Finally, identifying any irregular variations remaining in the data.

Visual Representation of Data Trends