Forecasting: Exponential Smoothing, MSE

1. Diagram
   Observation

   Table header shows 'Week' with rows 1 through 7.

Week

Time period index for the sales data.

Discrete integers from 1 to 7.

1. Diagram
   Observation

   Table column 'Sales' contains values 39, 44, 40, 45, 38, 43, 39.

Sales

Actual sales values collected over a seven-week period.

Real numbers representing sales quantities.

1. Diagram
   Observation

   Table column 'Forecast' is initially empty and later filled with calculated values.

Forecast

Predicted sales values for each week.

Real numbers representing forecasted quantities.

1. Audio
   Observation

   The actual sales will be represented by $A_t$.

2. Diagram
   Observation

   Label '$A_t$' appears above the 'Sales' column.

$A_t$

Actual value at time t.

Real numbers.

1. Audio
   Observation

   Note that some authors use $Y_t$ to represent actual values.

2. Diagram
   Observation

   Text box states 'Actual Value $A_t$ or $Y_t$'.

$Y_t$

Alternative notation for the actual value at time t.

Real numbers.

1. Audio
   Observation

   Forecast values on the other hand are generally represented by $F_t$.

2. Diagram
   Observation

   Label '$F_t$' appears above the 'Forecast' column.

$F_t$

Forecast value for time t.

Real numbers.

1. Audio
   Observation

   where alpha is a value between zero and one referred to as the smoothing constant.

2. Diagram
   Observation

   Text box states 'Smoothing Constant 0 <= alpha <= 1'.

$\alpha$

Smoothing constant used in the exponential smoothing formula.

Real numbers such that $0 \le \alpha \le 1$.

1. Formula
   Observation

   $α = 0.2$

$\alpha$

Smoothing constant, a weight applied to the most recent actual observation.

$0 < \alpha < 1$

1. Formula
   Observation

   $A_t$ in the formula $F_{t+1} = \alpha A_t + (1 - \alpha)F_t$

$A_t$

Actual value (sales) at time period t.

Real numbers

1. Formula
   Observation

   $F_t$ in the formula $F_{t+1} = \alpha A_t + (1 - \alpha)F_t$

$F_t$

Forecasted value for time period t.

Real numbers

1. Audio
   Observation

   mean squared error MSE

2. Formula
   Observation

   MSE = 11.58

MSE

Mean Squared Error, a measure of the average of the squares of the errors.

Non-negative real numbers

1. Audio
   Observation

   The exponential smoothing method uses the formula $F_{t+1} = F_t + \alpha(A_t - F_t)$.

2. Formula
   Observation

   Orange box displays $F_{t+1} = F_t + \alpha(A_t - F_t)$.

The first form of the exponential smoothing formula calculates the next period's forecast by adding a fraction of the previous period's forecast error to the previous period's forecast.

1. $\alpha$ is the smoothing constant, $0 \le \alpha \le 1$.

2. $A_t$ is the actual value at time t.

3. $F_t$ is the forecast value at time t.

1. $A_t$
2. $F_t$
3. $\alpha$

1. Audio
   Observation

   To make calculations easier, this formula can be rewritten as $F_{t+1} = \alpha A_t + (1 - \alpha)F_t$.

2. Formula
   Observation

   Blue box displays $F_{t+1} = \alpha A_t + (1 - \alpha)F_t$.

The second form of the formula expresses the new forecast as a weighted average of the last actual value and the last forecast value. The video notes this form is easier for calculations.

1. $\alpha$ is the smoothing constant, $0 \le \alpha \le 1$.

2. $A_t$ is the actual value at time t.

3. $F_t$ is the forecast value at time t.

1. Exponential Smoothing Formula (Error Correction Form)
2. $A_t$
3. $F_t$
4. $\alpha$

1. Audio
   Observation

   If no forecast value for the first period is given, then we assume $F_1 = A_1$.

2. Formula
   Observation

   Yellow highlight in the Forecast column for Week 1 shows $F_1 = A_1$.

When starting exponential smoothing without a prior forecast, the standard assumption is to set the first forecast equal to the first actual value.

1. No initial forecast value is provided for the first period.

1. $A_t$
2. $F_t$

1. Formula
   Observation

   $F_{t+1} = \alpha A_t + (1 - \alpha)F_t$

The forecast for the next period is a weighted average of the actual value of the current period and the forecast of the current period.

1. Requires an initial forecast value, often set to the first actual observation.

2. The smoothing constant $\alpha$ must be between 0 and 1.

1. Audio
   Observation

   To obtain the mean squared error MSE, we first obtain the forecast errors, square them, and then find the mean or average of the squared errors.

MSE is a metric used to evaluate the accuracy of a forecasting model. It is calculated by taking the average of the squared differences between the actual values and the forecasted values.

1. Errors are typically not calculated for the first period if the forecast is initialized with the first actual value.

1. Exponential Smoothing Formula

1. Audio
   Observation

   To make calculations easier, this formula can be rewritten as... Note that both formulas will give the same result.

2. Diagram
   Observation

   A double-headed arrow connects the orange and blue formula boxes.

1. Expression

   $$F_{t+1} = F_t + \alpha(A_t - F_t)$$
   Explanation

   Start with the error correction form of the formula.

   Justification

   Given in the video.

   Shown in the video
2. Expression

   $$F_{t+1} = F_t + \alpha A_t - \alpha F_t$$
   Explanation

   Distribute the smoothing constant alpha into the parentheses.

   Justification

   Algebraic distribution.

   Derived from the video
3. Expression

   $$F_{t+1} = \alpha A_t + F_t - \alpha F_t$$
   Explanation

   Rearrange the terms to group the $F_t$ terms together.

   Justification

   Commutative property of addition.

   Derived from the video
4. Expression

   $$F_{t+1} = \alpha A_t + (1 - \alpha)F_t$$
   Explanation

   Factor out $F_t$ from the grouped terms.

   Justification

   Factoring.

   Derived from the video

The two forms of the exponential smoothing formula are algebraically equivalent, but the weighted average form is computationally simpler.

1. Audio
   Observation

   0.2 times 40 plus 0.8 times 40, which gives 40. For F5, we have 0.2 times 45 plus 0.8 times 40, which gives 41...

2. Formula
   Observation

   $F_4 = 0.2(40) + 0.8(40.00)$ $F_5 = 0.2(45) + 0.8(40.00)$ $F_6 = 0.2(38) + 0.8(41.00)$ $F_7 = 0.2(43) + 0.8(40.40)$ $F_8 = 0.2(39) + 0.8(40.92)$

1. Expression

   $$F_4 = 0.2(40) + 0.8(40.00) = 40.00$$
   Explanation

   Calculate the forecast for week 4 using the actual sales from week 3 and the forecast for week 3.

   Justification

   Applying the exponential smoothing formula with $\alpha = 0.2$.

   Shown in the video
2. Expression

   $$F_5 = 0.2(45) + 0.8(40.00) = 41.00$$
   Explanation

   Calculate the forecast for week 5 using the actual sales from week 4 and the forecast for week 4.

   Justification

   Applying the exponential smoothing formula.

   Shown in the video
3. Expression

   $$F_6 = 0.2(38) + 0.8(41.00) = 40.40$$
   Explanation

   Calculate the forecast for week 6 using the actual sales from week 5 and the forecast for week 5.

   Justification

   Applying the exponential smoothing formula.

   Shown in the video
4. Expression

   $$F_7 = 0.2(43) + 0.8(40.40) = 40.92$$
   Explanation

   Calculate the forecast for week 7 using the actual sales from week 6 and the forecast for week 6.

   Justification

   Applying the exponential smoothing formula.

   Shown in the video
5. Expression

   $$F_8 = 0.2(39) + 0.8(40.92) = 40.54$$
   Explanation

   Calculate the forecast for week 8 using the actual sales from week 7 and the forecast for week 7.

   Justification

   Applying the exponential smoothing formula.

   Shown in the video

The forecasts for weeks 4 through 8 are calculated sequentially.

1. Audio
   Observation

   So the forecast error for week two is 44 minus 39, which is 5... Next we square the errors... Since we only have six periods with errors, we add up these errors and divide by six to obtain a mean squared error of 11.58.

2. Formula
   Observation

   Error = Sales - Forecast Error^2 MSE = 11.58

1. Expression

   $$E_2 = 44 - 39.00 = 5.00$$
   Explanation

   Calculate the forecast error for week 2.

   Justification

   Subtracting the forecast from the actual sales.

   Shown in the video
2. Expression

   $$E_3 = 40 - 40.00 = 0.00$$
   Explanation

   Calculate the forecast error for week 3.

   Justification

   Subtracting the forecast from the actual sales.

   Shown in the video
3. Expression

   $$E_4 = 45 - 40.00 = 5.00$$
   Explanation

   Calculate the forecast error for week 4.

   Justification

   Subtracting the forecast from the actual sales.

   Shown in the video
4. Expression

   $$E_5 = 38 - 41.00 = -3.00$$
   Explanation

   Calculate the forecast error for week 5.

   Justification

   Subtracting the forecast from the actual sales.

   Shown in the video
5. Expression

   $$E_6 = 43 - 40.40 = 2.60$$
   Explanation

   Calculate the forecast error for week 6.

   Justification

   Subtracting the forecast from the actual sales.

   Shown in the video
6. Expression

   $$E_7 = 39 - 40.92 = -1.92$$
   Explanation

   Calculate the forecast error for week 7.

   Justification

   Subtracting the forecast from the actual sales.

   Shown in the video
7. Expression

   $$E_2^2 = 5.00^2 = 25.00$$
   Explanation

   Square the error for week 2.

   Justification

   Squaring the calculated errors.

   Shown in the video
8. Expression

   $$E_3^2 = 0.00^2 = 0.00$$
   Explanation

   Square the error for week 3.

   Justification

   Squaring the calculated errors.

   Shown in the video
9. Expression

   $$E_4^2 = 5.00^2 = 25.00$$
   Explanation

   Square the error for week 4.

   Justification

   Squaring the calculated errors.

   Shown in the video
10. Expression

    $$E_5^2 = (-3.00)^2 = 9.00$$
    Explanation

    Square the error for week 5.

    Justification

    Squaring the calculated errors.

    Shown in the video
11. Expression

    $$E_6^2 = 2.60^2 = 6.76$$
    Explanation

    Square the error for week 6.

    Justification

    Squaring the calculated errors.

    Shown in the video
12. Expression

    $$E_7^2=(-1.92)^2=3.6864\approx3.69$$
    Explanation

    The exact squared error is 3.6864; the video displays it rounded as 3.69.

    Justification

    Squaring the calculated errors.

    Shown in the video
13. Expression

    $$MSE_{\mathrm{display}}=(25.00+0.00+25.00+9.00+6.76+3.69)/6=11.575\approx11.58$$
    Explanation

    Averaging the six displayed, individually rounded squared errors gives 11.575, reported on screen as 11.58.

    Justification

    Calculating the mean of the squared errors.

    Shown in the video
14. Expression

    $$MSE_{\mathrm{exact}}=(25+0+25+9+6.76+3.6864)/6=11.5744\approx11.57$$
    Explanation

    Keeping the last squared error unrounded until the final average gives 11.57 to two decimal places.

    Justification

    Independent arithmetic check of the same data.

    Derived from the video

The video reports MSE 11.58 from displayed rounded squared errors; using full precision until the end gives 11.57 to two decimals.

1. Audio
   Observation

   We then calculate $F_2$ as 0.2 times 39 plus 0.8 times 39... As a result, we just usually assume that $F_2$ equals $A_1$... This means that our first real use of the formula begins with $F_3$. So $F_3$ is going to be 0.2 times 44 plus 0.8 times 39, which gives 40.

2. Formula
   Observation

   Screen shows $F_2 = 0.2(39) + 0.8(39.00) = 39.00$, and $F_3 = 0.2(44) + 0.8(39.00) = 40.00$.

Calculate the forecasts for weeks 2 and 3 using exponential smoothing with $\alpha = 0.2$, given the sales data $A_1 = 39$ and $A_2 = 44$.

1. $\alpha = 0.2$

2. $A_1 = 39$

3. $A_2 = 44$

4. $F_1 = A_1 = 39$

Find $F_2$ and $F_3$.

1. Expression

   $$F_2 = 0.2(39) + 0.8(39) = 39$$
   Explanation

   Calculate $F_2$ using the formula and the assumed $F_1$.

   Justification

   Substitution into the weighted average formula.

   Shown in the video
2. Expression

   $$F_2 = A_1 = 39$$
   Explanation

   Recognize that calculating $F_2$ when $F_1 = A_1$ always yields $A_1$, so it can be directly copied.

   Justification

   Algebraic simplification noted in the video.

   Shown in the video
3. Expression

   $$F_3 = 0.2(44) + 0.8(39) = 8.8 + 31.2 = 40$$
   Explanation

   Calculate $F_3$ using the formula with $A_2$ and $F_2$.

   Justification

   Substitution into the weighted average formula.

   Shown in the video

$F_2 = 39$, $F_3 = 40$

The calculations match the values displayed in the table on screen.

1. Diagram
   Observation

   Table showing Week, Sales, Forecast, Error, and Error^2 columns being filled out.

Given weekly sales data and an initial forecast, calculate subsequent forecasts using exponential smoothing with $\alpha = 0.2$, and then compute the Mean Squared Error (MSE) for the model.

1. Weekly Sales data: Week 1: 39, Week 2: 44, Week 3: 40, Week 4: 45, Week 5: 38, Week 6: 43, Week 7: 39.

2. Initial Forecast: $F_2 = 39.00$ (which is $A_1$).

3. Smoothing constant: $\alpha = 0.2$.

Find the forecasts for weeks 3 through 8 and the MSE for the model based on weeks 2 through 7.

1. Expression

   $$F_{t+1} = 0.2(A_t) + 0.8(F_t)$$
   Explanation

   Apply the exponential smoothing formula iteratively to find forecasts for weeks 3 to 8.

   Justification

   Using the provided formula and $\alpha$ value.

   Shown in the video
2. Expression

   $$E_t = A_t - F_t$$
   Explanation

   Calculate the forecast error for each week from 2 to 7 by subtracting the forecast from the actual sales.

   Justification

   Definition of forecast error.

   Shown in the video
3. Expression

   $$E_t^2$$
   Explanation

   Square each of the calculated forecast errors.

   Justification

   Required step for MSE calculation.

   Shown in the video
4. Expression

   $$MSE = \frac{\sum E_t^2}{n}$$
   Explanation

   Sum the squared errors and divide by the number of error terms ($n=6$) to find the MSE.

   Justification

   Definition of Mean Squared Error.

   Shown in the video

Forecasts: $F_3=40.00, F_4=40.00, F_5=41.00, F_6=40.40, F_7=40.92, F_8=40.54$. MSE is 11.58 from the rounded table, or 11.57 if exact squared errors are averaged first.

The calculations are shown step-by-step in the video's table, and the final MSE is explicitly stated.

1. Animation
   Observation

   A table with columns Week, Sales, and Forecast is shown. The Forecast column is progressively filled with calculated values as the video explains the process.

1. Table with Week, Sales, and Forecast columns

2. Calculated forecast values

1. The Forecast column starts empty.

2. Values for $F_1$, $F_2$, and $F_3$ are sequentially added to the Forecast column.

1. The Week and Sales columns remain static throughout the segment.

The visual progression demonstrates the step-by-step application of the exponential smoothing formula to generate forecasts for each subsequent period.

1. Animation
   Observation

   A table is progressively filled with calculated values for Forecast, Error, and Error^2 columns.

1. Table with columns: Week, Sales, Forecast, Error, Error^2

2. Formulas on the right side of the screen

1. Values appear in the 'Forecast' column row by row.

2. Values appear in the 'Error' column row by row.

3. Values appear in the 'Error^2' column row by row.

4. The final MSE value appears at the bottom.

1. The 'Week' and 'Sales' columns remain static.

2. The general layout of the table and formulas remains unchanged.

The animation visually demonstrates the sequential nature of the calculations, where each new value depends on previously calculated ones.

1. Audio
   Observation

   As a result, we just usually assume that $F_2$ equals $A_1$ and not bother calculating it at all.

Students might waste time calculating $F_2$ using the full formula when $F_1$ is assumed to be $A_1$.

The video explicitly points out that if $F_1 = A_1$, then $F_2$ will always equal $A_1$, so it is more efficient to simply copy the value of $A_1$ to $F_2$.

1. Audio
   Observation

   When computing errors for exponential smoothing forecasts, we do not calculate an error for period one unless otherwise stated.

2. Diagram
   Observation

   A red 'X' is drawn over the Error cell for Week 1.

An error should be calculated for every period, including the first one.

In exponential smoothing, the first period's forecast is often initialized with the first period's actual value, making the error zero and uninformative. Therefore, errors are typically calculated starting from the second period.

1. Audio
   Observation

   Note that both formulas will give the same result, it is just a little easier to use the second one.

2. Diagram
   Observation

   Double-headed arrow between the two formula boxes.

The error correction form and the weighted average form of the exponential smoothing equation are mathematically equivalent, but the latter is preferred for manual calculation.

1. Audio
   Observation

   Next we compute the mean squared error.

The MSE is calculated using the forecast errors, which are derived from the forecasts generated by the exponential smoothing method.

1. Audio
   Observation

   To make calculations easier, this formula can be rewritten as...

1. Exponential Smoothing Formula (Error Correction Form)
2. Exponential Smoothing Formula (Weighted Average Form)
3. s0-cr-formula-equivalence

1. Audio
   Observation

   If no forecast value for the first period is given, then we assume $F_1$ equals $A_1$.

1. Initial Forecast Assumption

1. Formula
   Observation

   $F_{t+1} = \alpha A_t + (1 - \alpha)F_t$

1. Exponential Smoothing Formula

1. Audio
   Observation

   To obtain the mean squared error MSE, we first obtain the forecast errors, square them, and then find the mean or average of the squared errors.

1. Mean Squared Error (MSE)

1. Audio
   Observation

   we do not calculate an error for period one unless otherwise stated.

1. Calculating Error for the First Period